Mutation (Jordan algebra)

In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first introduced by Max Koecher in his Jordan algebraic approach to Hermitian symmetric spaces and bounded symmetric domains of tube type. Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compact subgroup. Both groups act transitively on the compactification. The theory has been extended to cover all Hermitian symmetric spaces using the theory of Jordan pairs or Jordan triple systems. Koecher obtained the results in the more general case directly from the Jordan algebra case using the fact that only Jordan pairs associated with period two automorphisms of Jordan algebras are required.
Definitions.
Let "A" be a unital Jordan algebra over a field "k" of characteristic ≠ 2. For "a" in "A" define the Jordan multiplication operator on "A" by
and the quadratic representation "Q"("a") by
It satisfies
the fundamental identity
the commutation or homotopy identity
where
In particular if "a" or "b" is invertible then
It follows that "A" with the operations "Q" and "R" and the identity element defines a quadratic Jordan algebra, where a "quadratic Jordan algebra" consists of a vector space "A" with a distinguished element 1 and a quadratic map of "A" into endomorphisms of "A", "a" ↦ "Q"("a"), satisfying the conditions:
The Jordan triple product is defined by
so that
There are also the formulas
For "y" in "A" the mutation "A""y" is defined to the vector space "A" with multiplication
If "Q"("y") is invertible, the mutual is called a proper mutation or isotope.
Quadratic Jordan algebras.
Let "A" be a quadratic Jordan algebra over a field "k" of characteristic ≠ 2. Following , a linear Jordan algebra structure can be associated with "A" such that, if "L"("a") is Jordan multiplication, then the quadratic structure is given by "Q"("a") = 2"L"("a")2 − "L"("a"2).
Firstly the axiom "Q"("a")"R"("b","a") = "R"("a","b")"Q"("a") can be strengthened to
Indeed, applied to "c", the first two terms give
Switching "b" and "c" then gives
Now let
Replacing "b" by "a" and "a" by 1 in the identity above gives
In particular
The Jordan product is given by
so that
The formula above shows that 1 is an identity. Defining "a"2 by "a"∘"a" = "Q"("a")1, the only remaining condition to be verified is the Jordan identity
In the fundamental identity
Replace "a" by "a" + "t"1, set "b" = 1 and compare the coefficients of "t"2 on both sides:
Setting "b" = 1 in the second axiom gives
and therefore "L"("a") must commute with "L"("a"2).
Inverses.
Let "A" be a unital Jordan algebra over a field "k" of characteristic ≠ 2. An element "a" in a unital Jordan algebra "A" is said to be invertible if there is an element "b" such that "ab" = 1 and "a"2"b" = "a".
Properties.
If and , then . The Jordan identity can be polarized by replacing by and taking the coefficient of . This gives
Taking or and or shows that commutes with and commutes with . Hence . Applying gives . Hence . Conversely if and , then the second relation gives . So both and are invertible. The first gives so that and are each other's inverses. Since commutes with it commutes with its inverse . Similarly commutes with . So and .
Indeed, if is invertible then the above implies is invertible with inverse . Any inverse "b" satisfies , so . Conversely if is invertible let . Then
. The fundamental identity then implies that and are each other's inverses so that .
This follows from the formula .
Suppose that . Then by the fundamental identity is invertible, so is invertible.
This is an immediate consequence of the fundamental identity and the fact that is invertible if and only and are invertible.
In the commutation identity , set with . Then and . Since commutes with , .
If and commute, then implies . Conversely suppose that is invertible with inverse . Then . Morevoer commutes with and hence its inverse . So it commutes with
The algebra is commutative and associative, so if is an inverse there and . Conversely leaves invariant. So if it is bijective on it is bijective there. Thus lies in .
Elementary properties of proper mutations.
In fact
multiplication in the algebra "A""y" is given by
so by definition is commutative. It follows that
with
If "e" satisfies , then taking "a" = 1 gives
Taking "a" = "e" gives
so that "L"("y") and "L"("e") commute. Hence "y" is invertible and "e" = "y"−1.
Now for "y" invertible set
Then
Moreover,
Finally
since
Hence
Thus is a unital quadratic Jordan algebra. It therefore corresponds to a linear Jordan algebra with the associated Jordan multiplication operator "M"("a") given by
This shows that the operators satisfy the Jordan identity so that the proper mutation or isotope is a unital Jordan algebra. The correspondence with quadratic Jordan algebras shows that its quadratic representation is given by .
Nonunital mutations.
The definition of mutation also applies to non-invertible elements "y". If "A" is finite-dimensional over R or C, invertible elements "a" in "A" are dense, since invertibility is equivalent to the condition that det "Q"("a") ≠ 0. So by continuity the Jordan identity for proper mutations implies the Jordan identity for arbitrary mutations. In general the Jordan identity can be deduced from Macdonald's theorem for Jordan algebras because it involves only two elements of the Jordan algebra. Alternatively, the Jordan identity can be deduced by realizing the mutation inside a unital quadratic algebra.
For "a" in "A" define a quadratic structure on "A"1 = "A" ⊕ "k" by
formula_37
It can then be verified that is a unital quadratic Jordan algebra. The unital Jordan algebra to which it corresponds has "A""y" as an ideal, so that in particular "A""y" satisfies the Jordan identity. The identities for a unital quadratic Jordan algebra follow from the following compatibility properties of the quadratic map and the squaring map
Hua's identity.
Let "A" be a unital Jordan algebra. If "a", "b" and "a" – "b" are invertible, then Hua's identity holds:
In particular if "x" and 1 – "x" are invertible, then so too is 1 – "x"−1 with
To prove the identity for "x", set . Then . Thus commutes with and . Since , it also commutes with and . Since , also commutes with and .
It follows that . Moreover, since . So commutes with and hence . Thus has inverse .
Now let be the mutation of "A" defined by "a". The identity element of is . Moreover, an invertible element "c" in "A" is also invertible in with inverse .
Let in . It is invertible in "A", as is . So by the special case of Hua's identity for "x" in
Bergman operator.
If "A" is a unital Jordan algebra, the Bergman operator is defined for "a", "b" in "A" by
If "a" is invertible then
while if "b" is invertible then
In fact if "a" is invertible
and similarly if "b" is invertible.
More generally the Bergman operator satisfies a version of the commutation or homotopy identity:
and a version of the fundamental identity:
There is also a third more technical identity:
Quasi-invertibility.
Let "A" be a finite-dimensional unital Jordan algebra over a field "k" of characteristic ≠ 2. For a pair with and invertible define

In this case the Bergman operator defines an invertible operator on "A" and
In fact
Moreover, by definition is invertible if and only if is invertible. In that case
Indeed,
The assumption that be invertible can be dropped since can be defined only supposing that the Bergman operator is invertible. The pair is then said to be quasi-invertible. In that case is defined by the formula
If is invertible, then for some . The fundamental identity implies that . So by finite-dimensionality is invertible. Thus is invertible if and only if
 is invertible and in this case
In fact
so the formula follows by applying to both sides.
As before is quasi-invertible if and only if is quasi-invertible; and in that case
If "k" = R or C, this would follow by continuity from the special case where and were invertible. In general the proof requires four identities for the Bergman operator:
In fact applying to the identity yields
The first identity follows by cancelling and . The second identity follows by similar cancellation in
The third identity follows by applying the second identity to an element "d" and then switching the roles of "c" and "d". The fourth follows because
In fact is quasi-invertible if and only if is quasi-invertible in the mutation . Since this mutation might not necessarily unital this means that when an identity is adjoint becomes invertible in . This condition can be expressed as follows without mentioning the mutation or homotope:
In fact if is quasi-invertible, then satisfies the first identity by definition. The second follows because . Conversely the conditions state that in the conditions imply that is the inverse of . On the other hand,
 for in . Hence is invertible.
Equivalence relation.
Let "A" be a finite-dimensional unital Jordan algebra over a field "k" of characteristic ≠ 2.
Two pairs with invertible are said to be equivalent if is invertible and .
This is an equivalence relation, since if is invertible so that a pair is equivalent to itself. It is symmetric since from the definition . It is transitive. For suppose that is a third pair with invertible and . From the above
is invertible and
As for quasi-invertibility, this definition can be extended to the case where and are not assumed to be invertible.
Two pairs are said to be equivalent if is quasi-invertible and . When "k" = R or C, the fact that this more general definition also gives an equivalence relation can deduced from the invertible case by continuity. For general "k", it can also be verified directly:
The equivalence class of is denoted by .
Structure groups.
Let be a finite-dimensional complex semisimple unital Jordan algebra. If is an operator on , let be its transpose with respect to the trace form. Thus
, , and . The structure group of "A" consists of "g" in such that
They form a group . The automorphism group Aut "A" of "A" consists of invertible complex linear operators "g" such that "L"("ga") = "gL"("a")"g"−1 and "g1" = 1. Since an automorphism "g" preserves the trace form, "g"−1 = "g""t".
The complex Jordan algebra "A" is the complexification of a real Euclidean Jordan algebra "E", for which the trace form defines an inner product. There is an associated involution on which gives rise to a complex inner product on . The unitary structure group Γ"u"("A") is the subgroup of Γ("A") consisting of unitary operators, so that . The identity component of is denoted by . It is a connected closed subgroup of .
The structure group Γ("A") acts naturally on "X". For "g" in Γ("A"), set
Then is quasi-invertible if and only if is quasi-invertible and
In fact the covariance relations for "g" with "Q" and the inverse imply that
if "x" is invertible and so everywhere by density. In turn this implies the relation for the quasi-inverse. If "a" is invertible then "Q"("a") lies in Γ("A") and if ("a","b") is quasi-invertible "B"("a","b") lies in Γ("A"). So both types of operators act on "X".
The defining relations for the structure group show that it is a closed subgroup of formula_56 of . Since , the corresponding complex Lie algebra contains the operators . The commutators span the complex Lie algebra of derivations of . The operators span formula_56 and
satisfy and
Geometric properties of quotient space.
Let "A" be a finite-dimensional complex unital Jordan algebra which is semisimple, i.e. the trace form Tr "L"("ab") is non-degenerate. Let be the quotient of by the equivalence relation. Let be the subset of "X" of classes . The map , is injective. A subset of is defined to be open if and only if is open for all .
The transition maps of the atlas with charts are given by
and are injective and holomorphic since
with derivative
This defines the structure of a complex manifold on "X" because on .

Indeed, all the polynomial functions are non-trivial since . Therefore, there is a such that for all "i", which is precisely the criterion for to lie in .
 uses the Bergman operators to construct an explicit biholomorphism between "X" and a closed smooth algebraic subvariety of complex projective space. This implies in particular that is compact. There is a more direct proof of compactness using symmetry groups.
Given a Jordan frame ("e""i") in "E", for every "a" in "A" there is a "k" in "U" = Γ"u"("A") such that
with (and if "a" is invertible).
In fact, if ("a","b") is in "X" then it is equivalent to "k"("c","d") with "c" and "d" in the unital Jordan subalgebra , which is the complexification of .
Let be the complex manifold constructed for . Because is a direct sum of copies of C, "Z" is just a product of Riemann spheres, one for each . In particular it is compact. There is a natural map of "Z" into "X" which is continuous. Let "Y" be the image of "Z". It is compact and therefore coincides with the closure of "Y"0 = "A""e" ⊂ "A" = "X"0. The set "U"⋅"Y" is the continuous image of the compact set "U" × "Y". It is therefore compact. On the other hand, "U"⋅"Y"0 = "X"0, so it contains a dense subset of "X" and must therefore coincide with "X". So "X" is compact.
The above argument shows that every ("a","b") in "X" is equivalent to "k"("c","d") with "c" and "d" in and "k" in
. The mapping of "Z" into "X" is in fact an embedding. This is a consequence of being quasi-invertible in if and only if it is quasi-invertible in . Indeed, if is injective on "A", its restriction to is also injective. Conversely, the two equations for the quasi-inverse in imply that it is also a quasi-inverse in .
Möbius transformations.
Let be a finite-dimensional complex semisimple unital Jordan algebra. The group SL(2,C) acts by Möbius transformation on the Riemann sphere C ∪ {∞}, the one-point compactification of C. If "g" in SL(2,C) is given by the matrix
then
There is a generalization of this action of SL(2,C) to "A" and its compactification "X". In order to define this action, note that SL(2,C) is generated by the three subgroups of lower and upper unitriangular matrices and the diagonal matrices. It is also generated by the lower (or upper) unitriangular matrices, the diagonal matrices and the matrix
The matrix "J" corresponds to the Möbius transformation and can be written
The Möbius transformations fixing ∞ are just the upper triangular matrices. If "g" does not fix ∞, it sends ∞ to a finite point "a". But then "g" can be composed with an upper unitriangular to send "a" to 0 and then with "J" to send 0 to infinity.
For an element of , the action of in SL(2,C) is defined by the same formula
This defines an element of provided that is invertible in . The action is thus defined everywhere on if "g" is upper triangular. On the other hand, the action on "X" is simple to define for lower triangular matrices.
In fact on the invertible elements in "A", the operator satisfies . To define a biholomorphism "j" on "X" such that , it is enough to define these for in some suitable orbit of Γ("A") or Γ"u"("A"). On the other hand, as indicated above, given a Jordan frame ("e""i") in "E", for every "a" in "A" there is a "k" in "U" = Γ"u"("A") such that with .
The computation of in the associative commutative algebra is straightforward since it is a direct product. For and , the Bergman operator on has determinant . In particular for some λ ≠ 0. So that is equivalent to . Let . On , for a dense set of , the pair is equivalent to with "b" invertible. Then is equivalent to . Since is holomorphic it follows that "j" has a unique continuous extension to "X" such that for in , the extension is holomorphic and for ,
The holomorphic transformations corresponding to upper unitriangular matrices can be defined using the fact that they are the conjugates by "J" of lower unitriangular matrices, for which the action is already known. A direct algebraic construction is given in .
This action of is compatible with inclusions. More generally if is a Jordan frame, there is an action of on "A""e" which extends to "A". If and , then and give the action of the product of the lower and upper unitriangular matrices. If is invertible, the corresponding product of diagonal matrices act as . In particular the diagonal matrices give an action of and .
Holomorphic symmetry group.
Let be a finite-dimensional complex semisimple unital Jordan algebra. There is a transitive holomorphic action of a complex matrix group on the compact complex manifold . described analogously to in terms of generators and relations. acts on the corresponding finite-dimensional Lie algebra of holomorphic vector fields restricted to , so that is realized as a closed matrix group. It is the complexification of a compact Lie group without center, so a semisimple algebraic group. The identity component of the compact group acts transitively on , so that can be identified as a Hermitian symmetric space of compact type.
The group "G" is generated by three types of holomorphic transformation on :
The operators normalize the group of operators . Similarly the operator normalizes the structure group, . The operators also form a group of holomorphic transformations isomorphic to the additive group of . They generalize the upper unitriangular subgroup of . This group is normalized by the operators "W" of the structure group. The operator acts on as .
If is a scalar the operators and coincide with the operators corresponding to lower and upper unitriangular matrices in . Accordingly, there is a relation and is a subgroup of "G". defines the operators in terms of the flow associated to a holomorphic vector field on , while give a direct algebraic description.
Indeed, .
Let and be the complex Abelian groups formed by the symmetries and respectively. Let .
The two expressions for are equivalent as follows by conjugating by .
For invertible, Hua's identity can be rewritten
Moreover, and
The convariance relations show that the elements of fall into sets
The first expression for follows once it is established that no new elements appear in the fourth or subsequent sets. For this it suffices to show that
For then if there are three or more occurrences of , the number can be recursively reduced to two. Given in , pick so that and are invertible. Then
which lies in .
It suffices to check that if , then . If so , so .
Exchange relations.
For invertible, Hua's identity can be rewritten
Since , the operators belong to the group generated by .
For quasi-invertible pairs , there are the "exchange relations"
This identity shows that is in the group generated by . Taking inverses, it is equivalent to the identity .
To prove the exchange relations, it suffices to check that it valid when applied to points the dense set of points in for which is quasi-invertible. It then reduces to the identity:
In fact, if is quasi-invertible, then is quasi-invertible if and only if is quasi-invertible. This follows because is quasi-invertible if and only if is. Moreover, the above formula holds in this case.
For the proof, two more identities are required:
The first follows from a previous identity by applying the transpose. For the second, because of the transpose, it suffices to prove the first equality. Setting in the identity yields
so the identity follows by cancelling .
To prove the formula, the relations
and show that it is enough to prove that
Indeed, . On the other hand,
 and . So .
Now set . Then the exchange relations imply that lies in if and only if is quasi-invertible; and that lies in if and only if is in .
In fact if lies in , then is equivalent to , so it a quasi-invertible pair; the converse follows from the exchange relations. Clearly . The converse follows from and the criterion for to lie in .
Lie algebra of holomorphic vector fields.
The compact complex manifold is modelled on the space . The derivatives of the transition maps describe the tangent bundle through holomorphic transition functions . These are given by , so the structure group of the corresponding principal fiber bundle reduces to , the structure group of . The corresponding holomorphic vector bundle with fibre is the tangent bundle of the complex manifold . Its holomorphic sections are just holomorphic vector fields on "X". They can be determined directly using the fact that they must be invariant under the natural adjoint action of the known holomorphic symmetries of "X". They form a finite-dimensional complex semisimple Lie algebra. The restriction of these vector fields to "X"0 can be described explicitly. A direct consequence of this description is that the Lie algebra is three-graded and that the group of holomorphic symmetries of "X", described by generators and relations in and , is a complex linear semisimple algebraic group that coincides with the group of biholomorphisms of "X".
The Lie algebras of the three subgroups of holomorphic automorphisms of give rise to linear spaces of holomorphic vector fields on and hence .
Let
Then, defining formula_76 for , formula_77 forms a complex Lie algebra with
This gives the structure of a 3-graded Lie algebra. For elements in formula_77, the Lie bracket is given by
The group of Möbius transformations of "X" normalizes the Lie algebra formula_77. The transformation corresponding to the Weyl group element induces the involutive automorphism given by
More generally the action of a Möbius transformation
can be described explicitly. In terms of generators diagonal matrices act as
upper unitriangular matrices act as
and lower unitriangular matrices act as
This can be written uniformly in matrix notation as
In particular the grading corresponds to the action of the diagonal subgroup of , even with |α| = 1, so a copy of T.
The Killing form is given by
where is the symmetric bilinear form defined by
with the bilinear form corresponding to the trace form: .
More generally the generators of the group act by automorphisms on formula_77 as
The nondegeneracy of the Killing form is immediate from the explicit formula. By Cartan's criterion, formula_77 is semisimple. In the next section the group is realized as the complexification of a connected compact Lie group with trivial center, so semisimple. This gives a direct means to verify semisimplicity. The group "H" also acts transitively on "X".
To prove that formula_77 exhausts the holomorphic vector fields on , note the group T acts on holomorphic vector fields. The restriction of such a vector field to gives a holomorphic map of "A" into "A". The power series expansion around 0 is a convergent sum of homogeneous parts of degree . The action of scales the part of degree by . By taking Fourier coefficients with respect to T, the part of degree "m" is also a holomorphic vector field. Since conjugation by gives the inverse on , it follows that the only possible degrees are 0, 1 and 2. Degree 0 is accounted for by the constant fields. Since conjugation by interchanges degree 0 and degree 2, it follows that formula_97 account for all these holomorphic vector fields. Any further holomorphic vector field would have to appear in degree 1 and so would have the form for some in . Conjugation by "J" would give another such map "N". Moreover, . But then
Set and . Then
It follows that lies in for all and hence that lies in formula_56. So formula_77 is exactly the space of holomorphic vector fields on "X".
Compact real form.
Suppose acts trivially on formula_77. Then must leave the subalgebra invariant. Hence so must . This forces , so that . But then must leave the subalgebra invariant, so that and . If acts trivially, .
The group can thus be identified with its image in GL formula_77.
Let be the complexification of a Euclidean Jordan algebra . For , set . The trace form on defines a complex inner product on and hence an adjoint operation. The unitary structure group consists of those in that are in , i.e. satisfy . It is a closed subgroup of "U"("A"). Its Lie algebra consists of the skew-adjoint elements in formula_56. Define a conjugate linear involution on formula_77 by
This is a period 2 conjugate-linear automorphism of the Lie algebra. It induces an automorphism of , which on the generators is given by
Let be the fixed point subgroup of in . Let formula_108 be the fixed point subalgebra of in formula_77. Define a sesquilinear form on formula_77 by . This defines a complex inner product on formula_77 which restricts to a real inner product on formula_108. Both are preserved by . Let be the identity component of . It lies in . Let be the diagonal torus associated with a Jordan frame in "E". The action of is compatible with which sends a unimodular matrix formula_113 to formula_114. In particular this gives a homomorphism of into .
Now every matrix in can be written as a product
The factor in the middle gives another maximal torus in obtained by conjugating by . If with |α"i"| = 1, then gives the action of the diagonal torus and corresponds to an element of . The element lies in and its image is a Möbius transformation lying in . Thus is another torus in and coincides with the image of .
Since for the compact complex manifold corresponding to , if follows that , where is the image of . On the other hand, , so that
. On the other hand, the stabilizer of in is , since the fixed point subgroup of under is . Hence . In particular "H" is compact and connected since both "K" and "S" are. Because it is a closed subgroup of U formula_77, it is a Lie group. It contains "K" and hence its Lie algebra contains the operators with . It contains the image of and hence the elements with in . Since and , it follows that the Lie algebra formula_117 of contains for all in . Thus it contains formula_108.
They are equal because all skew-adjoint derivations of formula_108 are inner. In fact, since normalizes formula_108 and the action by conjugation is faithful, the map of formula_117 into the Lie algebra formula_122 of derivations of formula_108 is faithful. In particular formula_108 has trivial center. To show that formula_108 equals formula_117, it suffices to show that formula_122 coincides with formula_108. Derivations on formula_108 are skew-adjoint for the inner product given by minus the Killing form. Take the invariant inner product on formula_122 given by . Since formula_108 is invariant under formula_122 so is its orthogonal complement. They are both ideals in formula_122, so the Lie bracket between them must vanish. But then any derivation in the orthogonal complement would have 0 Lie bracket with formula_108, so must be zero. Hence formula_108 is the Lie algebra of . (This also follows from a dimension count since .)
The formulas above for the action of and show that the image of is closed in GL formula_77. Since acts transitively on and the stabilizer of in is , it follows that . The compactness of and closedness of implies that is closed in GL formula_77.
 is a closed subgroup of GL formula_77 so a real Lie group. Since it contains with or , its Lie algebra contains formula_77. Since formula_77 is the complexification of formula_108, like formula_108 all its derivations are inner and it has trivial center. Since the Lie algebra of normalizes formula_77 and o is the only element centralizing formula_77, as in the compact case the Lie algebra of must be formula_77. (This can also be seen by a dimension count since .) Since it is a complex subspace, is a complex Lie group. It is connected because it is the continuous image of the connected set .
Since formula_77 is the complexification of formula_108, is the complexification of .
Noncompact real form.
For in the spectral norm ||"a"|| is defined to be if with and in . It is independent of choices and defines a norm on . Let be the set of with ||"a"|| < 1 and let be the identity component of the closed subgroup of "G" carrying onto itself. It is generated by , the Möbius transformations in and the image of corresponding to a Jordan frame. Let τ be the conjugate-linear period 2 automorphism of formula_77 defined by
Let formula_150 be the fixed point algebra of τ. It is the Lie algebra of . It induces a period 2 automorphism of with fixed point subgroup . The group acts transitively on . The stabilizer of
0 is .
The noncompact real semisimple Lie group acts on "X" with an open orbit . As with the action of on the Riemann sphere, it has only finitely many orbits. This orbit structure can be explicitly described when the Jordan algebra is simple. Let be the subset of consisting of elements with exactly of the α"i" less than one and exactly of them greater than one. Thus . These sets are the intersections of the orbits of with . The orbits with are open. There is a unique compact orbit . It is the Shilov boundary "S" of "D" consisting of elements with in , the underlying Euclidean Jordan algebra. is in the closure of if and only if and .
In particular is in the closure of every orbit.
Jordan algebras with involution.
The preceding theory describes irreducible Hermitian symmetric spaces of tube type in terms of unital Jordan algebras. In general Hermitian symmetric spaces are described by a systematic extension of the above theory to Jordan pairs. In the development of , however, irreducible Hermitian symmetric spaces not of tube type are described in terms of period two automorphisms of simple Euclidean Jordan algebras. In fact any period 2 automorphism defines a Jordan pair: the general results of on Jordan pairs can be specialized to that setting.
Let τ be a period two automorphism of a simple Euclidean Jordan algebra "E" with complexification "A". There are corresponding decompositions "E" = "E"+ ⊕ "E"− and "A" = "A"+ ⊕ "A"− into ±1 eigenspaces of τ. Let . τ is assumed to satisfy the additional condition that the trace form on defines an inner product. For in , define to be the restriction of to "V". For a pair in , define and to be the restriction of and to . Then is simple if and only if the only subspaces invariant under all the operators and are and .
The conditions for quasi-invertibility in show that is invertible if and only if is invertible. The quasi-inverse is the same whether computed in or . A space of equivalence classes can be defined on pairs . It is a closed subspace of , so compact. It also has the structure of a complex manifold, modelled on . The structure group can be defined in terms of and it has as a subgroup the unitary structure group with identity component . The group is the identity component of the fixed point subgroup of τ in . Let be the group of biholomorphisms of generated by in , the identity component of , and the Abelian groups
 consisting of the and consisting of the with
 and in . It acts transitively on with stabilizer and
. The Lie algebra formula_151 of holomorphic vector fields on is a 3-graded Lie algebra,
Restricted to the components are generated as before by the constant functions into , by the operators and by the operators . The Lie brackets are given by exactly the same formula as before.
The spectral decomposition in and is accomplished using tripotents, i.e. elements such that . In this case is an idempotent in . There is a Pierce decomposition into eigenspaces of . The operators and
 commute, so leaves the eigenspaces above invariant.
In fact acts as 0 on , as 1/4 on and 1 on . This induces a Pierce decomposition . The subspace becomes a Euclidean Jordan algebra with unit under the mutation Jordan product }.
Two tripotents and are said to be orthogonal if all the operators when "a" and "b" are powers of and and if the corresponding idempotents and are orthogonal. In this case and generate a commutative associative algebra and , since . Let be in . Let be the finite-dimensional real subspace spanned by odd powers of . The commuting self-adjoint operators with odd powers of act on , so can be simultaneously diagonalized by an orthonormal basis . Since is a positive multiple of , rescaling if necessary, can be chosen to be a tripotent. They form an orthogonal family by construction. Since is in , it can be written with real. These are called the eigenvalues of (with respect to τ). Any other tripotent in has the form with , so the are up to sign the minimal tripotents in .
A maximal family of orthogonal tripotents in is called a Jordan frame. The tripotents are necessarily minimal. All Jordan frames have the same number of elements, called the rank of . Any two frames are related by an element in the subgroup of the structure group of preserving the trace form. For a given Jordan frame , any element in can be written in the form with and an operator in . The spectral norm of is defined by ||"a"|| = sup α"i" and is independent of choices. Its square equals the operator norm of . Thus becomes a complex normed space with open unit ball .
Note that for in , the operator is self-adjoint so that the norm |||| = ||||"n". Since , it follows that |||| = ||||"n". In particular the spectral norm of in is the square root of the spectral norm of . It follows that the spectral norm of is the same whether calculated in or . Since preserves both norms, the spectral norm on is obtained by restricting the spectral norm on .
For a Jordan frame , let . There is an action of on which extends to "V". If and , then and give the action of the product of the lower and upper unitriangular matrices. If with , then the corresponding product of diagonal matrices act as , where . In particular the diagonal matrices give an action of and .
As in the case without an automorphism τ, there is an automorphism θ of . The same arguments show that the fixed point subgroup is generated by and the image of . It is a compact connected Lie group. It acts transitively on ; the stabilizer of is . Thus , a Hermitian symmetric space of compact type.
Let be the identity component of the closed subgroup of carrying onto itself. It is generated by and the image of corresponding to a Jordan frame. Let ρ be the conjugate-linear period 2 automorphism of formula_151 defined by
Let formula_155 be the fixed point algebra of ρ. It is the Lie algebra of . It induces a period 2 automorphism of with fixed point subgroup . The group acts transitively on . The stabilizer of
0 is . is the Hermitian symmetric space of noncompact type dual to .
The Hermitian symmetric space of non-compact type have an unbounded realization, analogous the upper half-plane in C. Möbius transformations in corresponding to the Cayley transform and its inverse give biholomorphisms of the Riemann sphere exchanging the unit disk and the upper halfplane. When the Hermitian symmetric space is of tube type the same Möbius transformations map the disk in onto the tube domain were is the open self-dual convex cone of squares in the Euclidean Jordan algebra .
For Hermitian symmetric space not of tube type there is no action of on "X", so no analogous Cayley transform. A partial Cayley transform can be defined in that case for any given maximal tripotent in . It takes the disk in onto a Siegel domain of the second kind.
In this case is a Euclidean Jordan algebra and there is symmetric -valued bilinear form on such that the corresponding quadratic form takes values in its positive cone . The Siegel domain consists of pairs such that lies in .
The quadratic form on and the squaring operation on are given by
. The positive cone corresponds to with invertible.
Examples.
For simple Euclidean Jordan algebras with complexication , the Hermitian symmetric spaces of compact type can be described explicitly as follows, using Cartan's classification.
Type In. "A" is the Jordan algebra of "n" × "n" complex matrices with the operator Jordan product . It is the complexification of , the Euclidean Jordan algebra of self-adjoint "n" × "n" complex matrices. In this case acting on with formula_156 acting as . Indeed, this can be verified directly for diagonal, upper and lower unitriangular matrices which correspond to the operators , and . The subset corresponds to the matrices with invertible. In fact consider the space of linear maps from to . It is described by a pair (|) with in . This is a module for acting on the target space. There is also an action of induced by the action on the source space. The space of injective maps is invariant and acts freely on it. The quotient is the Grassmannian consisting of "n"-dimensional subspaces of . Define a map of into by sending to the injective map (|). This map induces an isomorphism of onto .
In fact let be an "n"-dimensional subspace of . If it is in general position, i.e. it and its orthogonal complement have trivial intersection with and
, it is the graph of an invertible operator .
So the image corresponds to (|) with and .
At the other extreme,
 and its orthogonal complement can be written as orthogonal sums , , where and are the intersections with and and with . Then and . Moreover, and . The subspace corresponds to the pair (|), where is the orthogonal projection of onto . So and .
The general case is a direct sum of these two cases. can be written as an orthogonal sum where and are the intersections with and
 and is their orthogonal complement in . Similarly the orthogonal complement of can be written .
Thus and , where are orthogonal complements. The direct sum is of the second kind and its orthogonal complement of the first.
Maps in the structure group correspond to in , with . The corresponding map on sends (|) to
(|). Similarly the map corresponding to sends (|) to
(|), the map corresponding to sends (|) to
(|) and the map corresponding to sends
(|) to (|). It follows that the map corresponding to sends
(|) to
On the other hand, if is invertible,
(|) is equivalent to
(|), whence the formula for the fractional linear transformation.
Type IIIn. "A" is the Jordan algebra of "n" × "n" symmetric complex matrices with the operator Jordan product . It is the complexification of , the Euclidean Jordan algebra of "n" × "n" symmetric real matrices. On , define a nondegenerate alternating bilinear form by . In matrix notation if formula_157,
Let denote the complex symplectic group, the subgroup of preserving ω. It consists of such that and is closed under . If formula_156 belongs to then
It has center }. In this case } acting on as . Indeed, this can be verified directly for diagonal, upper and lower unitriangular matrices which correspond to the operators , and . The subset corresponds to the matrices with invertible. In fact consider the space of linear maps from to . It is described by a pair (|) with in . This is a module for acting on the target space. There is also an action of induced by the action on the source space. The space of injective maps with isotropic image, i.e. ω vanishes on the image, is invariant. Moreover, acts freely on it. The quotient is the symplectic Grassmannian consisting of "n"-dimensional Lagrangian subspaces of . Define a map of into by sending to the injective map (|). This map induces an isomorphism of onto .
In fact let be an "n"-dimensional Lagrangian subspace of . Let be a Lagrangian subspace complementing . If they are in general position, i.e. they have trivial intersection with and
, than is the graph of an invertible operator with . So the image corresponds to (|) with and .
At the other extreme,
 and can be written as direct sums , , where and are the intersections with and and with . Then and . Moreover, and . The subspace corresponds to the pair (|), where is the projection of onto . Note that the pair (, ) is in duality with respect to ω and the identification between them induces the canonical symmetric bilinear form on . In particular "V"1 is identified with "U"2 and "V"2 with "U"1. Moreover, they are "V"1 and "U"1 are orthogonal with respect to the symmetric bilinear form on (. Hence the idempotent satisfies . So and lie in and is the image of (|).
The general case is a direct sum of these two cases. can be written as a direct sum where and are the intersections with and
 and is a complement in . Similarly can be written .
Thus and , where are complements. The direct sum is of the second kind. It has a complement of the first kind.
Maps in the structure group correspond to in , with . The corresponding map on sends (|) to
(|). Similarly the map corresponding to sends (|) to
(|), the map corresponding to sends (|) to
(|) and the map corresponding to sends
(|) to (|). It follows that the map corresponding to sends
(|) to
On the other hand, if is invertible,
(|) is equivalent to
(|), whence the formula for the fractional linear transformation.
Type II2n. "A" is the Jordan algebra of 2"n" × 2"n" skew-symmetric complex matrices and Jordan product where the unit is given by formula_161. It is the complexification of , the Euclidean Jordan algebra of self-adjoint "n" × "n" matrices with entries in the quaternions. This is discussed in and .
Type IVn. "A" is the Jordan algebra with Jordan product . It is the complexication of the rank 2 Euclidean Jordan algebra defined by the same formulas but with real coefficients. This is discussed in .
Type VI. The complexified Albert algebra. This is discussed in , and .
The Hermitian symmetric spaces of compact type for simple Euclidean Jordan algebras with period two automorphism can be described explicitly as follows, using Cartan's classification.
Type Ip,q. Let "F" be the space of "q" by "p" matrices over R with "p" ≠ "q". This corresponds to the automorphism of "E" = "H""p" + "q"(R) given by conjugating by the diagonal matrix with "p" diagonal entries equal to 1 and "q" to −1. Without loss of generality can be taken greater than . The structure is given by
the triple product . The space "X" can be identified with the Grassmannian of -dimensional subspace of . This has a natural embedding in by adding 0's in the last coordinates. Since any -dimensional subspace of can be represented in the form [|], the same is true for subspaces lying in . The last rows of must vanish and the mapping does not change if the last rows of are set equal to zero. So a similar representation holds for mappings, but now with "q" by "p" matrices. Exactly as when , it follows that there is an action of by fractional linear transformations.
Type IIn "F" is the space of real skew-symmetric "m" by "m" matrices. After removing a factor of , this corresponds to the period 2 automorphism given by complex conjugation on "E" = "H""n"(C).
Type V. "F" is the direct sum of two copies of the Cayley numbers, regarded as 1 by 2 matrices. This corresponds to the canonical period 2 automorphism defined by any minimal idempotent in "E" = "H"3(O).


Andrew Ogg

Andrew Pollard Ogg (born April 9, 1934, Bowling Green, Ohio) is an American mathematician, a professor emeritus of mathematics at the University of California, Berkeley.
Education.
Ogg was a student at Bowling Green State University in the mid 1950s. Ogg received his Ph.D. in 1961 from Harvard University under the supervision of John Tate.
Career.
Ogg worked in algebra and number theory. His accomplishments include the Grothendieck–Ogg–Shafarevich formula, Ogg's formula for the conductor of an elliptic curve, the Néron–Ogg–Shafarevich criterion and the 1975 characterization of supersingular primes, the starting point for the theory of monstrous moonshine. He also posed the torsion conjecture in 1973 and is the author of the book "Modular forms and Dirichlet series" (W. A. Benjamin, 1969).


Scale factor

A scale factor is usually a decimal which scales, or multiplies, some quantity. In the equation "y" = "Cx", "C" is the scale factor for "x". "C" is also the coefficient of "x", and may be called the constant of proportionality of "y" to "x". For example, doubling distances corresponds to a scale factor of two for distance, while cutting a cake in half results in pieces with a scale factor for volume of one half. The basic equation for it is image over preimage.
In the field of measurements, the scale factor of an instrument is sometimes referred to as sensitivity. The ratio of any two corresponding lengths in two similar geometric figures is also called a scale.


ENDLIBPATH (environment variable)


Topological category

In category theory, a discipline in mathematics, the notion of topological category has a number of different, inequivalent definitions.
In one approach, a topological category is a category that is enriched over the category of compactly generated Hausdorff spaces. They can be used as a foundation for higher category theory, where they can play the role of (formula_1,1)-categories. An important example of a topological category in this sense is given by the category of CW complexes, where each set Hom("X","Y") of continuous maps from "X" to "Y" is equipped with the compact-open topology.
In another approach, a topological category is defined as a category formula_2 along with a forgetful functor formula_3 that maps to the category of sets and has the following three properties:
An example of a topological category in this sense is the category of all topological spaces with continuous maps, where one uses the standard forgetful functor.


Terminal singularity


Pro-category


Discrete category

In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms:
Since by axioms, there is always the identity morphism between the same object, we can express the above as condition on the cardinality of the hom-set
Some authors prefer a weaker notion, where a discrete category merely needs to be equivalent to such a category.
Simple facts.
Any class of objects defines a discrete category when augmented with identity maps.
Any subcategory of a discrete category is discrete. Also, a category is discrete if and only if all of its subcategories are full.
The limit of any functor from a discrete category into another category is called a product, while the colimit is called a coproduct. Thus, for example, the discrete category with just two objects can be used as a diagram or diagonal functor to define a product or coproduct of two objects. Alternately, for a general category C and the discrete category 2, one can consider the functor category C2. The diagrams of 2 in this category are pairs of objects, and the limit of the diagram is the product.
The functor from Set to Cat that sends a set to the corresponding discrete category is left adjoint to the functor sending a small category to its set of objects. (For the right adjoint, see indiscrete category.)


Dodgson condensation

In mathematics, Dodgson condensation or method of contractants is a method of computing the determinants of square matrices. It is named for its inventor, Charles Lutwidge Dodgson (better known by his pseudonym, as Lewis Carroll, the popular author). The method in the case of an "n" × "n" matrix is to construct an ("n" − 1) × ("n" − 1) matrix, an ("n" − 2) × ("n" − 2), and so on, finishing with a 1 × 1 matrix, which has one entry, the determinant of the original matrix.
General method.
This algorithm can be described in the following four steps:
Examples.
Without zeros.
One wishes to find
All of the interior elements are non-zero, so there is no need to re-arrange the matrix.
We make a matrix of its 2 × 2 submatrices.
We then find another matrix of determinants:
We must then divide each element by the corresponding element of our original matrix. The interior of the original matrix is
formula_7, so after dividing we get
formula_8.
The process must be repeated to arrive at a 1 × 1 matrix.
formula_9
Dividing by the interior of the 3 × 3 matrix, which is just −5, gives formula_10 and −8 is indeed the determinant of the original matrix.
With zeros.
Simply writing out the matrices:
Here we run into trouble. If we continue the process, we will eventually be dividing by 0. We can perform four row exchanges on the initial matrix to preserve the determinant and repeat the process, with most of the determinants precalculated:
Hence, we arrive at a determinant of 36.
Desnanot–Jacobi identity and proof of correctness of the condensation algorithm.
The proof that the condensation method computes the determinant of the matrix if no divisions by zero are encountered is based on an identity known as the Desnanot–Jacobi identity (1841) or, more generally the Sylvester determinant identity (1851).
Let formula_13 be a square matrix, and for each formula_14, denote by formula_15 the matrix that results from formula_16 by deleting the formula_17-th row and the formula_18-th column. Similarly, for
formula_19, denote by formula_20 the matrix that results from formula_16 by deleting the formula_17-th and formula_18-th rows and the formula_24-th and formula_25-th columns.
Proof of the correctness of Dodgson condensation.
Rewrite the identity as
Now note that by induction it follows that when applying the Dodgson condensation procedure to a square matrix formula_28 of order formula_29, the matrix in the formula_30-th stage of the computation (where the first stage formula_31 corresponds to the matrix formula_28 itself) consists of all the "connected minors" of order formula_30
of formula_28, where a connected minor is the determinant of a connected formula_35 sub-block of adjacent entries of formula_28. In particular, in the last stage formula_37, one gets a matrix containing a single element equal to the unique connected minor of order formula_29, namely the determinant of formula_28.
Proof of the Desnanot-Jacobi identity.
We follow the treatment in Bressoud's book; for an alternative combinatorial proof see the paper by Zeilberger.
Denote formula_40 (up to sign, the formula_41-th minor of formula_16), and define a formula_35
matrix formula_44 by
to arrive at
where we use formula_51 to denote the formula_41-th entry of formula_16. The determinant of this matrix is formula_54.
Second, this is equal to the product of the determinants, formula_55. But clearly
formula_56
so the identity follows from equating the two expressions we obtained for formula_48 and dividing out by formula_58 (this is allowed if one thinks of the identities as polynomial identities over the ring of polynomials in the formula_59 indeterminate variables formula_60).


Quasi-projective variety

In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in scheme theory, where a "quasi-projective scheme" is a locally closed subscheme of some projective space.
Relationship to affine varieties.
An affine space is a Zariski-open subset of a projective space, and since any closed affine subset formula_1 can be expressed as an intersection of the projective completion formula_2 and the affine space embedded in the projective space, this implies that any affine variety is quasiprojective. There are locally closed subsets of projective space that are not affine, so that quasi-projective is more general than affine. Taking the complement of a single point in projective space of dimension at least 2 gives a non-affine quasi-projective variety. This is also an example of a quasi-projective variety that is neither affine nor projective.
Examples.
Since quasi-projective varieties generalize both affine and projective varieties, they are sometimes referred to simply as "varieties". Varieties isomorphic to affine algebraic varieties as quasi-projective varieties are called affine varieties; similarly for projective varieties. For example, the complement of a point in the affine line, i.e., formula_3, is isomorphic to the zero set of the polynomial formula_4 in the affine plane. As an affine set formula_5 is not closed since any polynomial zero on the complement must be zero on the affine line. For another example, the complement of any conic in projective space of dimension 2 is affine. Varieties isomorphic to open subsets of affine varieties are called quasi-affine.
Quasi-projective varieties are "locally affine" in the same sense that a manifold is locally Euclidean : every point of a quasi-projective variety has a neighborhood which is an affine variety. This yields a basis of affine sets for the Zariski topology on a quasi-projective variety.


Extranatural transformation

In mathematics, specifically in category theory, an extranatural transformation is a generalization of the notion of natural transformation.
Definition.
Let formula_1 and formula_2 two functors of categories.
A family formula_3 is said to be natural in "a" and extranatural in "b" and "c" if the following holds:
Properties.
Extranatural transformations can be used to define wedges and thereby ends (dually co-wedges and co-ends), by setting formula_13 (dually formula_14) constant.
Extranatural transformations can be defined in terms of dinatural transformations, of which they are a special case.


Sylow theorems

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.
For a prime number formula_1, a Sylow "p"-subgroup (sometimes "p"-Sylow subgroup) of a group formula_2 is a maximal formula_1-subgroup of formula_2, i.e., a subgroup of formula_2 that is a "p"-group (so that the order of every group element is a power of formula_1) that is not a proper subgroup of any other formula_1-subgroup of formula_2. The set of all Sylow formula_1-subgroups for a given prime formula_1 is sometimes written formula_11.
The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group formula_2 the order (number of elements) of every subgroup of formula_2 divides the order of formula_2. The Sylow theorems state that for every prime factor "formula_1" of the order of a finite group formula_2, there exists a Sylow formula_1-subgroup of formula_2 of order formula_19, the highest power of formula_1 that divides the order of formula_2. Moreover, every subgroup of order "formula_19" is a Sylow "formula_1"-subgroup of formula_2, and the Sylow formula_1-subgroups of a group (for a given prime formula_1) are conjugate to each other. Furthermore, the number of Sylow formula_1-subgroups of a group for a given prime formula_1 is congruent to formula_29.
Theorems.
Motivation.
The Sylow theorems are a powerful statement about the structure of groups in general, but is also powerful in applications of finite group theory. This is because they give a method for using the prime decomposition of the cardinality of a finite group formula_2 to give statements about the structure of its subgroups: essentially, it gives a technique to transport basic number-theoretic information about a group to its group structure. From this observation, classifying finite groups becomes a game of finding which combinations/constructions of groups of smaller order can be applied to construct a group. For example, a typical application of these theorems is in the classification of finite groups of some fixed cardinality, e.g. formula_31.
Statement.
Collections of subgroups that are each maximal in one sense or another are common in group theory. The surprising result here is that in the case of formula_32, all members are actually isomorphic to each other and have the largest possible order: if formula_33 with formula_34 where does not divide , then every Sylow -subgroup has order formula_35. That is, is a -group and formula_36. These properties can be exploited to further analyze the structure of .
The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in "Mathematische Annalen".
The following weaker version of theorem 1 was first proved by Augustin-Louis Cauchy, and is known as Cauchy's theorem.
Consequences.
The Sylow theorems imply that for a prime number formula_1 every Sylow formula_1-subgroup is of the same order, formula_19. Conversely, if a subgroup has order formula_19, then it is a Sylow formula_1-subgroup, and so is isomorphic to every other Sylow formula_1-subgroup. Due to the maximality condition, if formula_43 is any formula_1-subgroup of formula_2, then formula_43 is a subgroup of a formula_1-subgroup of order formula_19.
A very important consequence of Theorem 2 is that the condition formula_49 is equivalent to saying that the Sylow formula_1-subgroup of formula_2 is a normal subgroup. However, there are groups that have normal subgroups but no normal Sylow subgroups, such as formula_52.
Sylow theorems for infinite groups.
There is an analogue of the Sylow theorems for infinite groups. One defines a Sylow -subgroup in an infinite group to be a "p"-subgroup (that is, every element in it has -power order) that is maximal for inclusion among all -subgroups in the group. Such subgroups exist by Zorn's lemma. Let formula_53 denote the set of conjugacy classes of a subgroup formula_54
Examples.
A simple illustration of Sylow subgroups and the Sylow theorems are the dihedral group of the "n"-gon, "D"2"n". For "n" odd, 2 = 21 is the highest power of 2 dividing the order, and thus subgroups of order 2 are Sylow subgroups. These are the groups generated by a reflection, of which there are "n", and they are all conjugate under rotations; geometrically the axes of symmetry pass through a vertex and a side.
By contrast, if "n" is even, then 4 divides the order of the group, and the subgroups of order 2 are no longer Sylow subgroups, and in fact they fall into two conjugacy classes, geometrically according to whether they pass through two vertices or two faces. These are related by an outer automorphism, which can be represented by rotation through π/"n", half the minimal rotation in the dihedral group.
Another example are the Sylow p-subgroups of "GL"2("F""q"), where "p" and "q" are primes ≥ 3 and , which are all abelian. The order of "GL"2("F""q") is . Since , the order of . Thus by Theorem 1, the order of the Sylow "p"-subgroups is "p"2"n".
One such subgroup "P", is the set of diagonal matrices formula_55, "x" is any primitive root of "F""q". Since the order of "F""q" is , its primitive roots have order "q" − 1, which implies that or "x""m" and all its powers have an order which is a power of "p". So, "P" is a subgroup where all its elements have orders which are powers of "p". There are "pn" choices for both "a" and "b", making |"P"| = "p"2"n". This means "P" is a Sylow "p"-subgroup, which is abelian, as all diagonal matrices commute, and because Theorem 2 states that all Sylow "p"-subgroups are conjugate to each other, the Sylow "p"-subgroups of "GL"2("F""q") are all abelian.
Example applications.
Since Sylow's theorem ensures the existence of p-subgroups of a finite group, it's worthwhile to study groups of prime power order more closely. Most of the examples use Sylow's theorem to prove that a group of a particular order is not simple. For groups of small order, the congruence condition of Sylow's theorem is often sufficient to force the existence of a normal subgroup.
Cyclic group orders.
Some non-prime numbers "n" are such that every group of order "n" is cyclic. One can show that "n" = 15 is such a number using the Sylow theorems: Let "G" be a group of order 15 = 3 · 5 and "n"3 be the number of Sylow 3-subgroups. Then "n"3 formula_56 5 and "n"3 ≡ 1 (mod 3). The only value satisfying these constraints is 1; therefore, there is only one subgroup of order 3, and it must be normal (since it has no distinct conjugates). Similarly, "n"5 must divide 3, and "n"5 must equal 1 (mod 5); thus it must also have a single normal subgroup of order 5. Since 3 and 5 are coprime, the intersection of these two subgroups is trivial, and so "G" must be the internal direct product of groups of order 3 and 5, that is the cyclic group of order 15. Thus, there is only one group of order 15 (up to isomorphism).
Small groups are not simple.
A more complex example involves the order of the smallest simple group that is not cyclic. Burnside's "pa qb" theorem states that if the order of a group is the product of one or two prime powers, then it is solvable, and so the group is not simple, or is of prime order and is cyclic. This rules out every group up to order 30 .
If "G" is simple, and |"G"| = 30, then "n"3 must divide 10 ( = 2 · 5), and "n"3 must equal 1 (mod 3). Therefore, "n"3 = 10, since neither 4 nor 7 divides 10, and if "n"3 = 1 then, as above, "G" would have a normal subgroup of order 3, and could not be simple. "G" then has 10 distinct cyclic subgroups of order 3, each of which has 2 elements of order 3 (plus the identity). This means "G" has at least 20 distinct elements of order 3.
As well, "n"5 = 6, since "n"5 must divide 6 ( = 2 · 3), and "n"5 must equal 1 (mod 5). So "G" also has 24 distinct elements of order 5. But the order of "G" is only 30, so a simple group of order 30 cannot exist.
Next, suppose |"G"| = 42 = 2 · 3 · 7. Here "n"7 must divide 6 ( = 2 · 3) and "n"7 must equal 1 (mod 7), so "n"7 = 1. So, as before, "G" can not be simple.
On the other hand, for |"G"| = 60 = 22 · 3 · 5, then "n"3 = 10 and "n"5 = 6 is perfectly possible. And in fact, the smallest simple non-cyclic group is "A"5, the alternating group over 5 elements. It has order 60, and has 24 cyclic permutations of order 5, and 20 of order 3.
Wilson's theorem.
Part of Wilson's theorem states that
for every prime "p". One may easily prove this theorem by Sylow's third theorem. Indeed,
observe that the number "np" of Sylow's "p"-subgroups
in the symmetric group "Sp" is . On the other hand, . Hence, . So, .
Fusion results.
Frattini's argument shows that a Sylow subgroup of a normal subgroup provides a factorization of a finite group. A slight generalization known as Burnside's fusion theorem states that if "G" is a finite group with Sylow "p"-subgroup "P" and two subsets "A" and "B" normalized by "P", then "A" and "B" are "G"-conjugate if and only if they are "NG"("P")-conjugate. The proof is a simple application of Sylow's theorem: If "B"="Ag", then the normalizer of "B" contains not only "P" but also "Pg" (since "Pg" is contained in the normalizer of "Ag"). By Sylow's theorem "P" and "Pg" are conjugate not only in "G", but in the normalizer of "B". Hence "gh"−1 normalizes "P" for some "h" that normalizes "B", and then "A""gh"−1 = "B"h−1 = "B", so that "A" and "B" are "NG"("P")-conjugate. Burnside's fusion theorem can be used to give a more powerful factorization called a semidirect product: if "G" is a finite group whose Sylow "p"-subgroup "P" is contained in the center of its normalizer, then "G" has a normal subgroup "K" of order coprime to "P", "G" = "PK" and "P"∩"K" = {1}, that is, "G" is "p"-nilpotent.
Less trivial applications of the Sylow theorems include the focal subgroup theorem, which studies the control a Sylow "p"-subgroup of the derived subgroup has on the structure of the entire group. This control is exploited at several stages of the classification of finite simple groups, and for instance defines the case divisions used in the Alperin–Brauer–Gorenstein theorem classifying finite simple groups whose Sylow 2-subgroup is a quasi-dihedral group. These rely on J. L. Alperin's strengthening of the conjugacy portion of Sylow's theorem to control what sorts of elements are used in the conjugation.
Proof of the Sylow theorems.
The Sylow theorems have been proved in a number of ways, and the history of the proofs themselves is the subject of many papers, including Waterhouse, Scharlau, Casadio and Zappa, Gow, and to some extent Meo.
One proof of the Sylow theorems exploits the notion of group action in various creative ways. The group acts on itself or on the set of its "p"-subgroups in various ways, and each such action can be exploited to prove one of the Sylow theorems. The following proofs are based on combinatorial arguments of Wielandt. In the following, we use formula_58 as notation for "a divides b" and formula_59 for the negation of this statement.
and no power of "p" remains in any of the factors inside the product on the right. Hence , completing the proof.
Algorithms.
The problem of finding a Sylow subgroup of a given group is an important problem in computational group theory.
One proof of the existence of Sylow "p"-subgroups is constructive: if "H" is a "p"-subgroup of "G" and the index ["G":"H"] is divisible by "p", then the normalizer "N" = "NG"("H") of "H" in "G" is also such that ["N" : "H"] is divisible by "p". In other words, a polycyclic generating system of a Sylow "p"-subgroup can be found by starting from any "p"-subgroup "H" (including the identity) and taking elements of "p"-power order contained in the normalizer of "H" but not in "H" itself. The algorithmic version of this (and many improvements) is described in textbook form in Butler, including the algorithm described in Cannon. These versions are still used in the GAP computer algebra system.
In permutation groups, it has been proven, in Kantor and Kantor and Taylor, that a Sylow "p"-subgroup and its normalizer can be found in polynomial time of the input (the degree of the group times the number of generators). These algorithms are described in textbook form in Seress, and are now becoming practical as the constructive recognition of finite simple groups becomes a reality. In particular, versions of this algorithm are used in the Magma computer algebra system.


Gordana Todorov

Gordana Todorov (born July 24, 1949) is a mathematician interested in noncommutative algebra, representation theory, Artin algebra, and cluster algebra. She is a professor of mathematics at Northeastern University.
Biography.
Todorov earned her Ph.D. in 1978, at Brandeis University. Her dissertation, "Almost Split Sequences in the Representation Theory of Certain Classes of Artin Algebras", was supervised by Maurice Auslander.
Todorov is married to mathematician Kiyoshi Igusa, with whom she is a frequent co-author. The Igusa–Todorov functions and Igusa–Todorov endomorphism algebras are named for their joint work. Todorov is also the namesake of Todorov's theorem on preprojective partitions, and the Gentle–Todorov theorem on abelian categories.


T. N. Venkataramana

Tyakal Nanjundiah Venkataramana (born 1958) is an Indian mathematician who specialises in algebraic groups and automorphic forms.
He was awarded the Shanti Swarup Bhatnagar Prize for Science and Technology in 2001, the highest science award in India, in the mathematical sciences category. Venkataramana's first major work was the extension of G. A. Margulis's work on arithmeticity of higher rank lattices to the case of groups in positive characteristics. He also has contributions to non-vanishing theorems on cohomology of arithmetic groups, to Lefschetz type theorems on restriction of cohomology on locally symmetric spaces and to arithmeticity of monodromy groups.


Least-squares function approximation

In mathematics, least squares function approximation applies the principle of least squares to function approximation, by means of a weighted sum of other functions. The best approximation can be defined as that which minimizes the difference between the original function and the approximation; for a least-squares approach the quality of the approximation is measured in terms of the squared differences between the two.
Functional analysis.
A generalization to approximation of a data set is the approximation of a function by a sum of other functions, usually an orthogonal set:
with the set of functions {formula_2} an orthonormal set over the interval of interest, : see also Fejér's theorem. The coefficients {formula_3} are selected to make the magnitude of the difference ||||2 as small as possible. For example, the magnitude, or norm, of a function over the can be defined by:
where the ‘*’ denotes complex conjugate in the case of complex functions. The extension of Pythagoras' theorem in this manner leads to function spaces and the notion of Lebesgue measure, an idea of “space” more general than the original basis of Euclidean geometry. The satisfy orthonormality relations:
where "δ""ij" is the Kronecker delta. Substituting function into these equations then leads to
the "n"-dimensional Pythagorean theorem:
The coefficients {"a""j"} making ||"f" − "f""n"||2 as small as possible are found to be:
The generalization of the "n"-dimensional Pythagorean theorem to "infinite-dimensional " real inner product spaces is known as Parseval's identity or Parseval's equation. Particular examples of such a representation of a function are the Fourier series and the generalized Fourier series.
Further discussion.
Using linear algebra.
It follows that one can find a "best" approximation of another function by minimizing the area between two functions, a continuous function formula_8 on formula_9 and a function formula_10 where formula_11 is a subspace of formula_12:
all within the subspace formula_11. Due to the frequent difficulty of evaluating integrands involving absolute value, one can instead define
as an adequate criterion for obtaining the least squares approximation, function formula_16, of formula_8 with respect to the inner product space formula_11.
As such, formula_19 or, equivalently, formula_20, can thus be written in vector form:
In other words, the least squares approximation of formula_8 is the function formula_23 closest to formula_8 in terms of the inner product formula_25. Furthermore, this can be applied with a theorem:


Dual Hahn polynomials

In mathematics, the dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined on a non-uniform lattice formula_1 and are defined as
for formula_3 and the parameters formula_4 are restricted to formula_5 is the falling and rising factorials, otherwise known as the Pochhammer symbol, and formula_6 is the generalized hypergeometric functions
 give a detailed list of their properties.
Orthogonality.
The Dual Hahn polynomials have the orthogonality condition
for formula_8. Where formula_9,
and
Numerical Instability.
As the value of formula_12 increases, the values that the discrete polynomials obtain also increases. As a result, to obtain numerical stability in calculating the polynomials you would use the renormalised Dual Hahn Polynomial as defined as
for formula_3.
Then the orthogonality condition becomes
for formula_8
Relation to other polynomials.
The Hahn polynomials, formula_17, is defined on the uniform lattice formula_18, and the parameters formula_4 are defined as formula_20. Then setting formula_21 the Hahn polynomials become the Tchebichef polynomials. Note that the Dual Hahn polynomials have a "q"-analog with an extra parameter "q" known as the Dual Hahn Q-polynomials
Racah polynomials are a generalization of dual Hahn polynomials


Hurwitz scheme

In algebraic geometry, the Hurwitz scheme formula_1 is the scheme parametrizing pairs (formula_2) where "C" is a smooth curve of genus "g" and has degree "d".


Modal matrix

In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.
Specifically the modal matrix formula_1 for the matrix formula_2 is the "n" × "n" matrix formed with the eigenvectors of formula_2 as columns in formula_1. It is utilized in the similarity transformation
where formula_6 is an "n" × "n" diagonal matrix with the eigenvalues of formula_2 on the main diagonal of formula_6 and zeros elsewhere. The matrix formula_6 is called the spectral matrix for formula_2. The eigenvalues must appear left to right, top to bottom in the same order as their corresponding eigenvectors are arranged left to right in formula_1.
Example.
The matrix
has eigenvalues and corresponding eigenvectors
A diagonal matrix formula_6, similar to formula_2 is
One possible choice for an invertible matrix formula_1 such that formula_5 is
Note that since eigenvectors themselves are not unique, and since the columns of both formula_1 and formula_6 may be interchanged, it follows that both formula_1 and formula_6 are not unique.
Generalized modal matrix.
Let formula_2 be an "n" × "n" matrix. A generalized modal matrix formula_1 for formula_2 is an "n" × "n" matrix whose columns, considered as vectors, form a canonical basis for formula_2 and appear in formula_1 according to the following rules:
One can show that
where formula_34 is a matrix in Jordan normal form. By premultiplying by formula_35, we obtain
Note that when computing these matrices, equation () is the easiest of the two equations to verify, since it does not require inverting a matrix.
Example.
This example illustrates a generalized modal matrix with four Jordan chains. Unfortunately, it is a little difficult to construct an interesting example of low order.
The matrix
has a single eigenvalue formula_37 with algebraic multiplicity formula_38. A canonical basis for formula_2 will consist of one linearly independent generalized eigenvector of rank 3 (generalized eigenvector rank; see generalized eigenvector), two of rank 2 and four of rank 1; or equivalently, one chain of three vectors formula_40, one chain of two vectors formula_41, and two chains of one vector formula_42, formula_43.
An "almost diagonal" matrix formula_34 in "Jordan normal form", similar to formula_2 is obtained as follows:
where formula_1 is a generalized modal matrix for formula_2, the columns of formula_1 are a canonical basis for formula_2, and formula_52. Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both formula_1 and formula_34 may be interchanged, it follows that both formula_1 and formula_34 are not unique.


Weyl module

In algebra, a Weyl module is a representation of a reductive algebraic group, introduced by and named after Hermann Weyl. In characteristic 0 these representations are irreducible, but in positive characteristic they can be reducible, and their decomposition into irreducible components can be hard to determine.


Noetherian ring

In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; that is, given any increasing sequence of left (or right) ideals:
there exists a natural number "n" such that:
Noetherian rings are named after Emmy Noether.
The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Lasker–Noether theorem, the Krull intersection theorem, and Hilbert's basis theorem hold for them. Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on "prime ideals". This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension.
Characterizations.
For noncommutative rings, it is necessary to distinguish between three very similar concepts:
For commutative rings, all three concepts coincide, but in general they are different. There are rings that are left-Noetherian and not right-Noetherian, and vice versa.
There are other, equivalent, definitions for a ring "R" to be left-Noetherian:
Similar results hold for right-Noetherian rings.
The following condition is also an equivalent condition for a ring "R" to be left-Noetherian and it is Hilbert's original formulation:
For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated.
Examples.
Rings that are not Noetherian tend to be (in some sense) very large. Here are some examples of non-Noetherian rings:
However, a non-Noetherian ring can be a subring of a Noetherian ring. Since any integral domain is a subring of a field, any integral domain that is not Noetherian provides an example. To give a less trivial example,
Indeed, there are rings that are right Noetherian, but not left Noetherian, so that one must be careful in measuring the "size" of a ring this way. For example, if "L" is a subgroup of Q2 isomorphic to Z, let "R" be the ring of homomorphisms "f" from Q2 to itself satisfying "f"("L") ⊂ "L". Choosing a basis, we can describe the same ring "R" as
This ring is right Noetherian, but not left Noetherian; the subset "I"⊂"R" consisting of elements with "a" = 0 and "γ" = 0 is a left ideal that is not finitely generated as a left "R"-module.
If "R" is a commutative subring of a left Noetherian ring "S", and "S" is finitely generated as a left "R"-module, then "R" is Noetherian. (In the special case when "S" is commutative, this is known as Eakin's theorem.) However this is not true if "R" is not commutative: the ring "R" of the previous paragraph is a subring of the left Noetherian ring "S" = Hom(Q2, Q2), and "S" is finitely generated as a left "R"-module, but "R" is not left Noetherian.
A unique factorization domain is not necessarily a Noetherian ring. It does satisfy a weaker condition: the ascending chain condition on principal ideals. A ring of polynomials in infinitely-many variables is an example of a non-Noetherian unique factorization domain.
A valuation ring is not Noetherian unless it is a principal ideal domain. It gives an example of a ring that arises naturally in algebraic geometry but is not Noetherian.
Key theorems.
Many important theorems in ring theory (especially the theory of commutative rings) rely on the assumptions that the rings are Noetherian.
Implication on injective modules.
Given a ring, there is a close connection between the behaviors of injective modules over the ring and whether the ring is a Noetherian ring or not. Namely, given a ring "R", the following are equivalent:
The endomorphism ring of an indecomposable injective module is local and thus Azumaya's theorem says that, over a left Noetherian ring, each indecomposable decomposition of an injective module is equivalent to one another (a variant of the Krull–Schmidt theorem).


Everett C. Dade

Everett Clarence Dade is a mathematician at University of Illinois at Urbana–Champaign working on finite groups and representation theory, who introduced the Dade isometry and Dade's conjecture.
Work.
The Dade isometry is an isometry from class functions on a subgroup "H" with support on a subset "K" of "H" to class functions on a group "G" . It was introduced by as a generalization and simplification of an isometry used by in their proof of the odd order theorem, and was used by in his revision of the character theory of the odd order theorem.
Dade's conjecture is a conjecture relating the numbers of characters of blocks of a finite group to the numbers of characters of blocks of local subgroups.


Tautological ring

In algebraic geometry, the tautological ring is the subring of the Chow ring of the moduli space of curves generated by tautological classes. These are classes obtained from 1 by pushforward along various morphisms described below. The tautological cohomology ring is the image of the tautological ring under the cycle map (from the Chow ring to the cohomology ring).
Definition.
Let formula_1 be the moduli stack of stable marked curves formula_2, such that
The last condition requires formula_3 in other words ("g","n") is not among (0,0), (0,1), (0,2), (1,0). The stack formula_1 then has dimension formula_5. Besides permutations of the marked points, the following morphisms between these moduli stacks play an important role in defining tautological classes:
The tautological rings formula_9 are simultaneously defined as the smallest subrings of the Chow rings closed under pushforward by forgetful and gluing maps.
The tautological cohomology ring formula_10 is the image of formula_11 under the cycle map. As of 2016, it is not known whether the tautological and tautological cohomology rings are isomorphic.
Generating set.
For formula_12 we define the class formula_13 as follows. Let formula_14 be the pushforward of 1 along the gluing map formula_15 which identifies the marked point "xk" of the first curve to one of the three marked points "yi" on the sphere (the latter choice is unimportant thanks to automorphisms). For definiteness order the resulting points as "x"1, ..., "x""k"−1, "y"1, "y"2, "x""k"+1, ..., "xn". Then formula_16 is defined as the pushforward of formula_17 along the forgetful map that forgets the point "y"2. This class coincides with the first Chern class of a certain line bundle.
For formula_18 we also define formula_19 be the pushforward of formula_20 along the forgetful map formula_21 that forgets the "k"-th point. This is independent of "k" (simply permute points).
These pushforwards of monomials (hereafter called basic classes) do not form a basis. The set of relations is not fully known.
Faber conjectures.
The tautological ring formula_25 on the moduli space of smooth "n"-pointed genus "g" curves simply consists of restrictions of classes in formula_9. We omit "n" when it is zero (when there is no marked point).
In the case formula_27 of curves with no marked point, Mumford conjectured, and Madsen and Weiss proved, that for any formula_28 the map formula_29 is an isomorphism in degree "d" for large enough "g". In this case all classes are tautological.
Although formula_34 trivially vanishes for formula_35 because of the dimension of formula_36, the conjectured bound is much lower. The conjecture would completely determine the structure of the ring: a polynomial in the formula_37 of cohomological degree "d" vanishes if and only if its pairing with all polynomials of cohomological degree formula_38 vanishes.
Parts (1) and (2) of the conjecture were proven. Part (3), also called the Gorenstein conjecture, was only checked for formula_39. For formula_40 and higher genus, several methods of constructing relations between formula_24 classes find the same set of relations which suggest that the dimensions of formula_34 and formula_43 are different. If the set of relations found by these methods is complete then the Gorenstein conjecture is wrong. Besides Faber's original non-systematic computer search based on classical maps between vector bundles over formula_44, the "d"-th fiber power of the universal curve formula_45, the following methods have been used to find relations:
These four methods are proven to give the same set of relations.
Similar conjectures were formulated for moduli spaces formula_1 of stable curves and formula_49 of compact-type stable curves. However, Petersen-Tommasi proved that formula_50 and formula_51 fail to obey the (analogous) Gorenstein conjecture. On the other hand, Tavakol proved that for genus "2" the moduli space of rational-tails stable curves formula_52 obeys the Gorenstein condition for every "n".


Hyperstructure

Hyperstructures are algebraic structures equipped with at least one multi-valued operation, called a "hyperoperation". The largest classes of the hyperstructures are the ones called formula_1 – structures.
A hyperoperation formula_2 on a nonempty set formula_3 is a mapping from formula_4 to the nonempty power set formula_5, meaning the set of all nonempty subsets of formula_3, i.e.
For formula_9 we define
formula_13 is a "semihypergroup" if formula_2 is an associative hyperoperation, i.e. formula_15 for all formula_16
Furthermore, a hypergroup is a semihypergroup formula_13, where the reproduction axiom is valid, i.e.
formula_18 for all formula_19


Nilpotent cone

In mathematics, the nilpotent cone formula_1 of a finite-dimensional semisimple Lie algebra formula_2 is the set of elements that act nilpotently in all representations of formula_3 In other words,
The nilpotent cone is an irreducible subvariety of formula_2 (considered as a vector space).
Example.
The nilpotent cone of formula_6, the Lie algebra of 2×2 matrices with vanishing trace, is the variety of all 2×2 traceless matrices with rank less than or equal to formula_7


Tilting theory

In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra.
Tilting theory was motivated by the introduction of reflection functors by ; these functors were used to relate representations of two quivers. These functors were reformulated by , and generalized by who introduced tilting functors. defined tilted algebras and tilting modules as further generalizations of this.
Definitions.
Suppose that "A" is a finite-dimensional unital associative algebra over some field. A finitely-generated right "A"-module "T" is called a tilting module if it has the following three properties:
Given such a tilting module, we define the endomorphism algebra "B" = End"A"("T"). This is another finite-dimensional algebra, and "T" is a finitely-generated left "B"-module.
The tilting functors Hom"A"("T",−), Ext("T",−), −⊗"B""T" and Tor(−,"T") relate the category mod-"A" of finitely-generated right "A"-modules to the category mod-"B" of finitely-generated right "B"-modules.
In practice one often considers hereditary finite dimensional algebras "A" because the module categories over such algebras are fairly well understood. The endomorphism algebra of a tilting module over a hereditary finite dimensional algebra is called a tilted algebra.
Facts.
Suppose "A" is a finite-dimensional algebra, "T" is a tilting module over "A", and "B" = End"A"("T"). Write "F"=Hom"A"("T",−), "F′"=Ext("T",−), "G"=−⊗"B""T", and "G′"=Tor(−,"T"). "F" is right adjoint to "G" and "F′" is right adjoint to "G′".
 showed that tilting functors give equivalences between certain subcategories of mod-"A" and mod-"B". Specifically, if we define the two subcategories formula_1 and formula_2 of "A"-mod, and the two subcategories formula_3 and formula_4 of "B"-mod, then formula_5 is a torsion pair in "A"-mod (i.e. formula_6 and formula_7 are maximal subcategories with the property formula_8; this implies that every "M" in "A"-mod admits a natural short exact sequence formula_9 with "U" in formula_6 and "V" in formula_7) and formula_12 is a torsion pair in "B"-mod. Further, the restrictions of the functors "F" and "G" yield inverse equivalences between formula_6 and formula_14, while the restrictions of "F′" and "G′" yield inverse equivalences between formula_7 and formula_16. (Note that these equivalences switch the order of the torsion pairs formula_5 and formula_12.)
Tilting theory may be seen as a generalization of Morita equivalence which is recovered if "T" is a projective generator; in that case formula_19 and formula_20.
If "A" has finite global dimension, then "B" also has finite global dimension, and the difference of "F" and "F"' induces an isometry between the Grothendieck groups K0("A") and K0("B").
In case "A" is hereditary (i.e. "B" is a tilted algebra), the global dimension of "B" is at most 2, and the torsion pair formula_12 splits, i.e. every indecomposable object of "B"-mod is either in formula_16 or in formula_14.
 and showed that in general "A" and "B" are derived equivalent (i.e. the derived categories Db("A"-mod) and Db("B"-mod) are equivalent as triangulated categories).
Generalizations and extensions.
A generalized tilting module over the finite-dimensional algebra "A" is a right "A"-module "T" with the following three properties:
These generalized tilting modules also yield derived equivalences between "A" and "B", where "B"=End"A"("T").
 extended the results on derived equivalence by proving that two finite-dimensional algebras "R" and "S" are derived equivalent if and only if "S" is the endomorphism algebra of a "tilting complex" over "R". Tilting complexes are generalizations of generalized tilting modules. A version of this theorem is valid for arbitrary rings "R" and "S".
 defined tilting objects in hereditary abelian categories in which all Hom- and Ext-spaces are finite-dimensional over some algebraically closed field "k". The endomorphism algebras of these tilting objects are the quasi-tilted algebras, a generalization of tilted algebras. The quasi-tilted algebras over "k" are precisely the finite-dimensional algebras over "k" of global dimension ≤ 2 such that every indecomposable module either has projective dimension ≤ 1 or injective dimension ≤ 1. classified the hereditary abelian categories that can appear in the above construction.
 defined tilting objects "T" in an arbitrary abelian category "C"; their definition requires that "C" contain the direct sums of arbitrary (possibly infinite) numbers of copies of "T", so this is not a direct generalization of the finite-dimensional situation considered above. Given such a tilting object with endomorphism ring "R", they establish tilting functors that provide equivalences between a torsion pair in "C" and a torsion pair in "R"-Mod, the category of "all" "R"-modules.
From the theory of cluster algebras came the definition of cluster category (from ) and cluster tilted algebra () associated to a hereditary algebra "A". A cluster tilted algebra arises from a tilted algebra as a certain semidirect product, and the cluster category of "A" summarizes all the module categories of cluster tilted algebras arising from "A".


Morphism

In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.
In category theory, "morphism" is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism.
The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the "objects" are simply "sets with some additional structure", and "morphisms" are "structure-preserving functions". In category theory, morphisms are sometimes also called arrows.
Definition.
A category "C" consists of two classes, one of and the other of . There are two objects that are associated to every morphism, the and the . A morphism "f" with source "X" and target "Y" is written "f" : "X" → "Y", and is represented diagrammatically by an from "X" to "Y".
For many common categories, objects are sets (often with some additional structure) and morphisms are functions from an object to another object. Therefore, the source and the target of a morphism are often called and respectively.
Morphisms are equipped with a partial binary operation, called . The composition of two morphisms "f" and "g" is defined precisely when the target of "f" is the source of "g", and is denoted "g" ∘ "f" (or sometimes simply "gf"). The source of "g" ∘ "f" is the source of "f", and the target of "g" ∘ "f" is the target of "g". The composition satisfies two axioms:
For a concrete category (a category in which the objects are sets, possibly with additional structure, and the morphisms are structure-preserving functions), the identity morphism is just the identity function, and composition is just ordinary composition of functions.
The composition of morphisms is often represented by a commutative diagram. For example,
The collection of all morphisms from "X" to "Y" is denoted Hom"C"("X","Y") or simply Hom("X", "Y") and called the hom-set between "X" and "Y". Some authors write Mor"C"("X","Y"), Mor("X", "Y") or C("X", "Y"). Note that the term hom-set is something of a misnomer, as the collection of morphisms is not required to be a set. A category where Hom("X", "Y") is a set for all objects "X" and "Y" is called locally small.
Note that the domain and codomain are in fact part of the information determining a morphism. For example, in the category of sets, where morphisms are functions, two functions may be identical as sets of ordered pairs (may have the same range), while having different codomains. The two functions are distinct from the viewpoint of category theory. Thus many authors require that the hom-classes Hom("X", "Y") be disjoint. In practice, this is not a problem because if this disjointness does not hold, it can be assured by appending the domain and codomain to the morphisms (say, as the second and third components of an ordered triple).
Some special morphisms.
Monomorphisms and epimorphisms.
A morphism "f": "X" → "Y" is called a monomorphism if "f" ∘ "g"1 = "f" ∘ "g"2 implies "g"1 = "g"2 for all morphisms "g"1, "g"2: "Z" → "X". A monomorphism can be called a "mono" for short, and we can use "monic" as an adjective. A morphism "f" has a left inverse or is a split monomorphism if there is a morphism "g": "Y" → "X" such that "g" ∘ "f" id"X". Thus "f" ∘ "g": "Y" → "Y" is idempotent; that is, ("f" ∘ "g")2 "f" ∘ ("g" ∘ "f") ∘ "g" "f" ∘ "g". The left inverse "g" is also called a retraction of "f".
Morphisms with left inverses are always monomorphisms, but the converse is not true in general; a monomorphism may fail to have a left inverse. In concrete categories, a function that has a left inverse is injective. Thus in concrete categories, monomorphisms are often, but not always, injective. The condition of being an injection is stronger than that of being a monomorphism, but weaker than that of being a split monomorphism.
Dually to monomorphisms, a morphism "f": "X" → "Y" is called an epimorphism if "g"1 ∘ "f" = "g"2 ∘ "f" implies "g"1 = "g"2 for all morphisms "g"1, "g"2: "Y" → "Z". An epimorphism can be called an "epi" for short, and we can use "epic" as an adjective. A morphism "f" has a right inverse or is a split epimorphism if there is a morphism "g": "Y" → "X" such that "f" ∘ "g" id"Y". The right inverse "g" is also called a section of "f". Morphisms having a right inverse are always epimorphisms, but the converse is not true in general, as an epimorphism may fail to have a right inverse.
If a monomorphism "f" splits with left inverse "g", then "g" is a split epimorphism with right inverse "f". In concrete categories, a function that has a right inverse is surjective. Thus in concrete categories, epimorphisms are often, but not always, surjective. The condition of being a surjection is stronger than that of being an epimorphism, but weaker than that of being a split epimorphism. In the category of sets, the statement that every surjection has a section is equivalent to the axiom of choice.
A morphism that is both an epimorphism and a monomorphism is called a bimorphism.
Isomorphisms.
A morphism "f": "X" → "Y" is called an isomorphism if there exists a morphism "g": "Y" → "X" such that "f" ∘ "g" = id"Y" and "g" ∘ "f" = id"X". If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so "f" is an isomorphism, and "g" is called simply the inverse of "f". Inverse morphisms, if they exist, are unique. The inverse "g" is also an isomorphism, with inverse "f". Two objects with an isomorphism between them are said to be isomorphic or equivalent.
While every isomorphism is a bimorphism, a bimorphism is not necessarily an isomorphism. For example, in the category of commutative rings the inclusion Z → Q is a bimorphism that is not an isomorphism. However, any morphism that is both an epimorphism and a "split" monomorphism, or both a monomorphism and a "split" epimorphism, must be an isomorphism. A category, such as Set, in which every bimorphism is an isomorphism is known as a balanced category.
Endomorphisms and automorphisms.
A morphism "f": "X" → "X" (that is, a morphism with identical source and target) is an endomorphism of "X". A split endomorphism is an idempotent endomorphism "f" if "f" admits a decomposition "f" = "h" ∘ "g" with "g" ∘ "h" = id. In particular, the Karoubi envelope of a category splits every idempotent morphism.
An automorphism is a morphism that is both an endomorphism and an isomorphism. In every category, the automorphisms of an object always form a group, called the automorphism group of the object.
Examples.
For more examples, see the entry category theory.


Factorization system

In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory.
Definition.
A factorization system ("E", "M") for a category C consists of two classes of morphisms "E" and "M" of C such that:
"Remark:" formula_10 is a morphism from formula_11 to formula_12 in the arrow category.
Orthogonality.
Two morphisms formula_13 and formula_14 are said to be "orthogonal", denoted formula_15, if for every pair of morphisms formula_4 and formula_5 such that formula_18 there is a unique morphism formula_9 such that the diagram
commutes. This notion can be extended to define the orthogonals of sets of morphisms by
Since in a factorization system formula_22 contains all the isomorphisms, the condition (3) of the definition is equivalent to
"Proof:" In the previous diagram (3), take formula_25 (identity on the appropriate object) and formula_26.
Equivalent definition.
The pair formula_27 of classes of morphisms of C is a factorization system if and only if it satisfies the following conditions:
Weak factorization systems.
Suppose "e" and "m" are two morphisms in a category C. Then "e" has the "left lifting property" with respect to "m" (respectively "m" has the "right lifting property" with respect to "e") when for every pair of morphisms "u" and "v" such that "ve" = "mu" there is a morphism "w" such that the following diagram commutes. The difference with orthogonality is that "w" is not necessarily unique.
A weak factorization system ("E", "M") for a category C consists of two classes of morphisms "E" and "M" of C such that:
This notion leads to a succinct definition of model categories: a model category is a pair consisting of a category C and classes of (so-called) weak equivalences "W", fibrations "F" and cofibrations "C" so that
A model category is a complete and cocomplete category equipped with a model structure. A map is called a trivial fibration if it belongs to formula_38 and it is called a trivial cofibration if it belongs to formula_39 An object formula_40 is called fibrant and the morphism formula_41 to the terminal object is a fibration, and it is called cobrant if the morphism formula_42 from the initial object is a cofibration.