Eric Katz

Eric Katz is a mathematician working in combinatorial algebraic geometry and arithmetic geometry. He is currently an associate professor in the Department of Mathematics at Ohio State University.
In joint work with Karim Adiprasito and June Huh, he resolved the Heron–Rota–Welsh conjecture on the log-concavity of the characteristic polynomial of matroids. With Joseph Rabinoff and David Zureick-Brown, he has given bounds on rational and torsion points on curves.
Education.
Katz went to Beachwood High School, in Beachwood, Ohio, a suburb of Cleveland. After earning a B.S. in Mathematics from Ohio State University in 1999, he pursued graduate studies at Stanford University, obtaining his Ph.D. in 2004 with a thesis written under the direction of Yakov Eliashberg and Ravi Vakil.


Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space formula_1 together with an operation called the Lie bracket, an alternating bilinear map formula_2, that satisfies the Jacobi identity. The vector space formula_1 together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.
Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras.
In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.
An elementary example is the space of three dimensional vectors formula_4 with the bracket operation defined by the cross product formula_5 This is skew-symmetric since formula_6, and instead of associativity it satisfies the Jacobi identity:
This is the Lie algebra of the Lie group of rotations of space, and each vector formula_8 may be pictured as an infinitesimal rotation around the axis "v", with velocity equal to the magnitude of "v". The Lie bracket is a measure of the non-commutativity between two rotations: since a rotation commutes with itself, we have the alternating property formula_9.
History.
Lie algebras were introduced to study the concept of infinitesimal transformations by Marius Sophus Lie in the 1870s, and independently discovered by Wilhelm Killing in the 1880s. The name "Lie algebra" was given by Hermann Weyl in the 1930s; in older texts, the term "infinitesimal group" is used.
Definitions.
Definition of a Lie algebra.
A Lie algebra is a vector space formula_10 over some field together with a binary operation formula_11 called the Lie bracket satisfying the following axioms:
Using bilinearity to expand the Lie bracket formula_19 and using alternativity shows that formula_20 for all elements "x", "y" in formula_14, showing that bilinearity and alternativity together imply
It is customary to denote a Lie algebra by a lower-case fraktur letter such as formula_25. If a Lie algebra is associated with a Lie group, then the algebra is denoted by the fraktur version of the group: for example the Lie algebra of SU("n") is formula_26.
Generators and dimension.
Elements of a Lie algebra formula_14 are said to generate it if the smallest subalgebra containing these elements is formula_14 itself. The "dimension" of a Lie algebra is its dimension as a vector space over "F". The cardinality of a minimal generating set of a Lie algebra is always less than or equal to its dimension.
See the classification of low-dimensional real Lie algebras for other small examples.
Subalgebras, ideals and homomorphisms.
The Lie bracket is not required to be associative, meaning that formula_29 need not equal formula_30. However, it is flexible. Nonetheless, much of the terminology of associative rings and algebras is commonly applied to Lie algebras. A "Lie subalgebra" is a subspace formula_31 which is closed under the Lie bracket. An "ideal" formula_32 is a subalgebra satisfying the stronger condition:
A Lie algebra "homomorphism" is a linear map compatible with the respective Lie brackets:
As for associative rings, ideals are precisely the kernels of homomorphisms; given a Lie algebra formula_14 and an ideal formula_36 in it, one constructs the "factor algebra" or "quotient algebra" formula_37, and the first isomorphism theorem holds for Lie algebras.
Since the Lie bracket is a kind of infinitesimal commutator of the corresponding Lie group, we say that two elements formula_38 "commute" if their bracket vanishes: formula_39.
The centralizer subalgebra of a subset formula_40 is the set of elements commuting with "S": that is, formula_41. The centralizer of formula_14 itself is the "center" formula_43. Similarly, for a subspace "S", the normalizer subalgebra of "S" is formula_44. Equivalently, if "S" is a Lie subalgebra, formula_45 is the largest subalgebra such that formula_46 is an ideal of formula_45.
Examples.
For formula_48, the commutator of two elementsformula_49shows formula_50 is a subalgebra, but not an ideal. In fact, every one-dimensional linear subspace of a Lie algebra has an induced abelian Lie algebra structure, which is generally not an ideal. For any simple Lie algebra, all abelian Lie algebras can never be ideals.
Direct sum and semidirect product.
For two Lie algebras formula_51 and formula_52, their direct sum Lie algebra is the vector space
formula_53consisting of all pairs formula_54, with the operation
so that the copies of formula_56 commute with each other: formula_57 Let formula_14 be a Lie algebra and formula_59 an ideal of formula_14. If the canonical map formula_61 splits (i.e., admits a section), then formula_14 is said to be a semidirect product of formula_59 and formula_64, formula_65. See also semidirect sum of Lie algebras.
Levi's theorem says that a finite-dimensional Lie algebra is a semidirect product of its radical and the complementary subalgebra (Levi subalgebra).
Derivations.
A "derivation" on the Lie algebra formula_14 (or on any non-associative algebra) is a linear map formula_67 that obeys the Leibniz law, that is,
for all formula_38. The "inner derivation" associated to any formula_70 is the adjoint mapping formula_71 defined by formula_72. (This is a derivation as a consequence of the Jacobi identity.) The outer derivations are derivations which do not come from the adjoint representation of the Lie algebra. If formula_14 is semisimple, every derivation is inner.
The derivations form a vector space formula_74, which is a Lie subalgebra of formula_75; the bracket is commutator. The inner derivations form a Lie subalgebra of formula_74.
Examples.
For example, given a Lie algebra ideal formula_77 the adjoint representation formula_78 of formula_14 acts as outer derivations on formula_59 since formula_81 for any formula_82 and formula_83. For the Lie algebra formula_84 of upper triangular matrices in formula_85, it has an ideal formula_86 of strictly upper triangular matrices (where the only non-zero elements are above the diagonal of the matrix). For instance, the commutator of elements in formula_87 and formula_88 givesformula_89shows there exist outer derivations from formula_87 in formula_91.
Split Lie algebra.
Let "V" be a finite-dimensional vector space over a field "F", formula_92 the Lie algebra of linear transformations and formula_93 a Lie subalgebra. Then formula_14 is said to be split if the roots of the characteristic polynomials of all linear transformations in formula_14 are in the base field "F". More generally, a finite-dimensional Lie algebra formula_14 is said to be split if it has a Cartan subalgebra whose image under the adjoint representation formula_97 is a split Lie algebra. A split real form of a complex semisimple Lie algebra (cf. #Real form and complexification) is an example of a split real Lie algebra. See also split Lie algebra for further information.
Vector space basis.
For practical calculations, it is often convenient to choose an explicit vector space basis for the algebra. A common construction for this basis is sketched in the article structure constants.
Definition using category-theoretic notation.
Although the definitions above are sufficient for a conventional understanding of Lie algebras, once this is understood, additional insight can be gained by using notation common to category theory, that is, by defining a Lie algebra in terms of linear maps—that is, morphisms of the category of vector spaces—without considering individual elements. (In this section, the field over which the algebra is defined is supposed to be of characteristic different from two.)
For the category-theoretic definition of Lie algebras, two braiding isomorphisms are needed. If is a vector space, the "interchange isomorphism" formula_98 is defined by
The "cyclic-permutation braiding" formula_100 is defined as
where formula_102 is the identity morphism.
Equivalently, formula_103 is defined by
With this notation, a Lie algebra can be defined as an object formula_105 in the category of vector spaces together with a morphism
that satisfies the two morphism equalities
and
Examples.
Vector spaces.
Any vector space formula_109 endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called abelian, cf. below. Any one-dimensional Lie algebra over a field is abelian, by the alternating property of the Lie bracket.
Special matrices.
Two important subalgebras of formula_120 are:
Matrix Lie algebras.
A complex matrix group is a Lie group consisting of matrices, formula_127, where the multiplication of "G" is matrix multiplication. The corresponding Lie algebra formula_1 is the space of matrices which are tangent vectors to "G" inside the linear space formula_129: this consists of derivatives of smooth curves in "G" at the identity: formula_130The Lie bracket of formula_14 is given by the commutator of matrices, formula_132. Given the Lie algebra, one can recover the Lie group as the image of the matrix exponential mapping formula_133 defined by formula_134, which converges for every matrix formula_135: that is, formula_136.
The following are examples of Lie algebras of matrix Lie groups:
Two dimensions.
Since
for any natural number formula_156 and any formula_157, one sees that the resulting Lie group elements are upper triangular 2×2 matrices with unit lower diagonal:
Representations.
Definitions.
Given a vector space "V", let formula_92 denote the Lie algebra consisting of all linear endomorphisms of "V", with bracket given by formula_132. A "representation" of a Lie algebra formula_14 on "V" is a Lie algebra homomorphism
A representation is said to be "faithful" if its kernel is zero. Ado's theorem states that every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space.
Adjoint representation.
For any Lie algebra formula_14, we can define a representation
given by formula_177; it is a representation on the vector space formula_14 called the adjoint representation.
Goals of representation theory.
One important aspect of the study of Lie algebras (especially semisimple Lie algebras) is the study of their representations. (Indeed, most of the books listed in the references section devote a substantial fraction of their pages to representation theory.) Although Ado's theorem is an important result, the primary goal of representation theory is not to find a faithful representation of a given Lie algebra formula_14. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather the goal is to understand "all" possible representation of formula_14, up to the natural notion of equivalence. In the semisimple case over a field of characteristic zero, Weyl's theorem says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The irreducible representations, in turn, are classified by a theorem of the highest weight.
Representation theory in physics.
The representation theory of Lie algebras plays an important role in various parts of theoretical physics. There, one considers operators on the space of states that satisfy certain natural commutation relations. These commutation relations typically come from a symmetry of the problem—specifically, they are the relations of the Lie algebra of the relevant symmetry group. An example would be the angular momentum operators, whose commutation relations are those of the Lie algebra formula_163 of the rotation group SO(3). Typically, the space of states is very far from being irreducible under the pertinent operators, but one can attempt to decompose it into irreducible pieces. In doing so, one needs to know the irreducible representations of the given Lie algebra. In the study of the quantum hydrogen atom, for example, quantum mechanics textbooks give (without calling it that) a classification of the irreducible representations of the Lie algebra formula_163.
Structure theory and classification.
Lie algebras can be classified to some extent. In particular, this has an application to the classification of Lie groups.
Abelian, nilpotent, and solvable.
Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras.
A Lie algebra formula_14 is "abelian" if the Lie bracket vanishes, i.e. ["x","y"] = 0, for all "x" and "y" in formula_14. Abelian Lie algebras correspond to commutative (or abelian) connected Lie groups such as vector spaces formula_185 or tori formula_186, and are all of the form formula_187 meaning an "n"-dimensional vector space with the trivial Lie bracket.
A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra formula_14 is "nilpotent" if the lower central series
Lie rings need not be Lie groups under addition. Any Lie algebra is an example of a Lie ring. Any associative ring can be made into a Lie ring by defining a bracket operator formula_113. Conversely to any Lie algebra there is a corresponding ring, called the universal enveloping algebra.
Lie rings are used in the study of finite p-groups through the "Lazard correspondence". The lower central factors of a "p"-group are finite abelian "p"-groups, so modules over Z/"p"Z. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the commutator of two coset representatives. The Lie ring structure is enriched with another module homomorphism, the "p"th power map, making the associated Lie ring a so-called restricted Lie ring.
Lie rings are also useful in the definition of a p-adic analytic groups and their endomorphisms by studying Lie algebras over rings of integers such as the p-adic integers. The definition of finite groups of Lie type due to Chevalley involves restricting from a Lie algebra over the complex numbers to a Lie algebra over the integers, and then reducing modulo "p" to get a Lie algebra over a finite field.


Rank of an abelian group

In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group "A" is the cardinality of a maximal linearly independent subset. The rank of "A" determines the size of the largest free abelian group contained in "A". If "A" is torsion-free then it embeds into a vector space over the rational numbers of dimension rank "A". For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved.
The term rank has a different meaning in the context of elementary abelian groups.
Definition.
A subset {"a""α"} of an abelian group is linearly independent (over Z) if the only linear combination of these elements that is equal to zero is trivial: if
where all but finitely many coefficients "n""α" are zero (so that the sum is, in effect, finite), then all summands are 0. Any two maximal linearly independent sets in "A" have the same cardinality, which is called the rank of "A".
The rank of an abelian group is analogous to the dimension of a vector space. The main difference with the case of vector space is a presence of torsion. An element of an abelian group "A" is classified as torsion if its order is finite. The set of all torsion elements is a subgroup, called the torsion subgroup and denoted "T"("A"). A group is called torsion-free if it has no non-trivial torsion elements. The factor-group "A"/"T"("A") is the unique maximal torsion-free quotient of "A" and its rank coincides with the rank of "A".
The notion of rank with analogous properties can be defined for modules over any integral domain, the case of abelian groups corresponding to modules over Z. For this, see finitely generated module#Generic rank.
Groups of higher rank.
Abelian groups of rank greater than 1 are sources of interesting examples. For instance, for every cardinal "d" there exist torsion-free abelian groups of rank "d" that are indecomposable, i.e. cannot be expressed as a direct sum of a pair of their proper subgroups. These examples demonstrate that torsion-free abelian group of rank greater than 1 cannot be simply built by direct sums from torsion-free abelian groups of rank 1, whose theory is well understood. Moreover, for every integer formula_4, there is a torsion-free abelian group of rank formula_5 that is simultaneously a sum of two indecomposable groups, and a sum of "n" indecomposable groups. Hence even the number of indecomposable summands of a group of an even rank greater or equal than 4 is not well-defined.
Another result about non-uniqueness of direct sum decompositions is due to A.L.S. Corner: given integers formula_6, there exists a torsion-free abelian group "A" of rank "n" such that for any partition formula_7 into "k" natural summands, the group "A" is the direct sum of "k" indecomposable subgroups of ranks formula_8. Thus the sequence of ranks of indecomposable summands in a certain direct sum decomposition of a torsion-free abelian group of finite rank is very far from being an invariant of "A".
Other surprising examples include torsion-free rank 2 groups "A""n","m" and "B""n","m" such that "A""n" is isomorphic to "B""n" if and only if "n" is divisible by "m".
For abelian groups of infinite rank, there is an example of a group "K" and a subgroup "G" such that
Generalization.
The notion of rank can be generalized for any module "M" over an integral domain "R", as the dimension over "R"0, the quotient field, of the tensor product of the module with the field:
It makes sense, since "R"0 is a field, and thus any module (or, to be more specific, vector space) over it is free.
It is a generalization, since any abelian group is a module over the integers. It easily follows that the dimension of the product over Q is the cardinality of maximal linearly independent subset, since for any torsion element x and any rational q


Probability semiring


P-recursive equation

In mathematics a P-recursive equation is a linear equation of sequences where the coefficient sequences can be represented as polynomials. P-recursive equations are linear recurrence equations (or linear recurrence relations or linear difference equations) with polynomial coefficients. These equations play an important role in different areas of mathematics, specifically in combinatorics. The sequences which are solutions of these equations are called holonomic, P-recursive or D-finite.
From the late 1980s on the first algorithms were developed to find solutions for these equations. Sergei A. Abramov, Marko Petkovšek and Mark van Hoeij described algorithms to find polynomial, rational, hypergeometric and d'Alembertian solutions.
Definition.
Let formula_1 be a field of characteristic zero (for example formula_2), formula_3 polynomials for formula_4,formula_5 a sequence and formula_6 an unknown sequence. The equationformula_7is called a linear recurrence equation with polynomial coefficients (all recurrence equations in this article are of this form). If formula_8 and formula_9 are both nonzero, then formula_10 is called the order of the equation. If formula_11 is zero the equation is called homogeneous, otherwise it is called inhomogeneous.
This can also be written as formula_12 where formula_13 is a linear recurrence operator with polynomial coefficients and formula_14 is the shift operator, i.e. formula_15.
Closed form solutions.
Let formula_16 or equivalently formula_17 be a recurrence equation with polynomial coefficients. There exist several algorithms which compute solutions of this equation. These algorithms can compute polynomial, rational, hypergeometric and d'Alembertian solutions. The solution of a homogeneous equation is given by the kernel of the linear recurrence operator: formula_18. As a subspace of the space of sequences this kernel has a basis. Let formula_19 be a basis of formula_20, then the formal sum formula_21 for arbitrary constants formula_22 is called the general solution of the homogeneous problem formula_23. If formula_24 is a particular solution of formula_17, i.e. formula_26, then formula_27 is also a solution of the inhomogeneous problem and it is called the general solution of the inhomogeneous problem.
Polynomial solutions.
In the late 1980s Sergei A. Abramov described an algorithm which finds the general polynomial solution of a recurrence equation, i.e. formula_28, with a polynomial right-hand sideformula_29. He (and a few years later Marko Petkovšek) gave a degree bound for polynomial solutions. This way the problem can simply be solved by considering a system of linear equations. In 1995 Abramov, Bronstein and Petkovšek showed that the polynomial case can be solved more efficiently by considering power series solution of the recurrence equation in a specific power basis (i.e. not the ordinary basis formula_30).
The other algorithms for finding more general solutions (e.g. rational or hypergeometric solutions) also rely on algorithms which compute polynomial solutions.
Rational solutions.
In 1989 Sergei A. Abramov showed that a general rational solution, i.e. formula_31, with polynomial right-hand side formula_32, can be found by using the notion of a universal denominator. A universal denominator is a polynomial formula_33 such that the denominator of every rational solution divides formula_33. Abramov showed how this universal denominator can be computed by only using the first and the last coefficient polynomial formula_8 and formula_9. Substituting this universal denominator for the unknown denominator of formula_37 all rational solutions can be found by computing all polynomial solutions of a transformed equation.
Hypergeometric solution.
A sequence formula_38 is called hypergeometric if the ratio of two consecutive terms is a rational function in formula_39, i.e. formula_40. This is the case if and only if the sequence is the solution of a first-order recurrence equation with polynomial coefficients. The set of hypergeometric sequences is not a subspace of the space of sequences as it is not closed under addition.
In 1992 Marko Petkovšek gave an algorithm to get the general hypergeometric solution of a recurrence equation where the right-hand side formula_41 is the sum of hypergeometric sequences. The algorithm makes use of the Gosper-Petkovšek normal-form of a rational function. With this specific representation it is again sufficient to consider polynomial solutions of a transformed equation.
A different and more efficient approach is due to Mark van Hoeij. Considering the roots of the first and the last coefficient polynomial formula_8 and formula_9 – called singularities – one can build a solution step by step making use of the fact that every hypergeometric sequence formula_44 has a representation of the formformula_45for some formula_46 with formula_47 for formula_48 and formula_49. Here formula_50 denotes the Gamma function and formula_51 the algebraic closure of the field formula_1. Then the formula_53 have to be singularities of the equation (i.e. roots of formula_8 or formula_9). Furthermore one can compute bounds for the exponents formula_56. For fixed values formula_57 it is possible to make an ansatz which gives candidates for formula_58. For a specific formula_58 one can again make an ansatz to get the rational function formula_60 by Abramov's algorithm. Considering all possibilities one gets the general solution of the recurrence equation.
D'Alembertian solutions.
A sequence formula_37 is called d'Alembertian if formula_62 for some hypergeometric sequences formula_63 and formula_64 means that formula_65 where formula_66 denotes the difference operator, i.e. formula_67. This is the case if and only if there are first-order linear recurrence operators formula_68 with rational coefficients such that formula_69.
1994 Abramov and Petkovšek described an algorithm which computes the general d'Alembertian solution of a recurrence equation. This algorithm computes hypergeometric solutions and reduces the order of the recurrence equation recursively.
Examples.
Signed permutation matrices.
The number of signed permutation matrices of size formula_70 can be described by the sequence formula_71. A signed permutation matrix is a square matrix which has exactly one nonzero entry in every row and in every column. The nonzero entries can be formula_72. The sequence is determined by the linear recurrence equation with polynomial coefficientsformula_73and the initial values formula_74. Applying an algorithm to find hypergeometric solutions one can find the general hypergeometric solutionformula_75for some constant formula_76. Also considering the initial values, the sequence formula_77 describes the number of signed permutation matrices.
Involutions.
The number of involutions formula_38 of a set with formula_79 elements is given by the recurrence equationformula_80Applying for example Petkovšek's algorithm it is possible to see that there is no polynomial, rational or hypergeometric solution for this recurrence equation.
Applications.
A function formula_81 is called hypergeometric if formula_82 where formula_83 denotes the rational functions in formula_79 and formula_85. A hypergeometric sum is a finite sum of the form formula_86 where formula_81 is hypergeometric. Zeilberger's creative telescoping algorithm can transform such a hypergeometric sum into a recurrence equation with polynomial coefficients. This equation can then be solved to get for example a linear combination of hypergeometric solutions which is called a closed form solution of formula_11.


Daniel Huybrechts

Daniel Huybrechts (9 November 1966) is a German mathematician, specializing in algebraic geometry.
Education and career.
Huybrechts studied mathematics from 1985 at the Humboldt University of Berlin, where in 1989 he earned his Diplom with Diplom thesis supervisor Herbert Kurke. In 1990–1992 Huybrechts studied at the Max Planck Institute for Mathematics in Bonn, where he earned his Ph.D. (Promotion) in 1992 under Herbert Kurke with thesis "Stabile Vektorbündel auf algebraischen Flächen. Tjurins Methode zum Studium der Geometrie der Modulräume". In the academic year 1994–1995 he was at the Institute for Advanced Study and in 1996 at IHES. In 1996 he was a research assistant at the University of Essen, where in 1998 he earned his Habilitierung. In 1997–1998 he was at the École normale supérieure. He was a professor in 1998–2002 at the University of Cologne and in 2002–2005 at the École polytechnique (Chargé de Cours) and, simultaneously, at the University of Paris VII. Since 2005 he has been a professor at the University of Bonn.
Huybrechts does research on K3 surfaces and their higher-dimensional analogues (compact hyperkähler manifolds) and moduli spaces of sheaves on varieties.
In 2010 he was an invited speaker at the International Congress of Mathematicians in Hyderabad and gave a talk "Hyperkähler Manifolds and Sheaves".


Complex conjugate representation

In mathematics, if is a group and is a representation of it over the complex vector space , then the complex conjugate representation is defined over the complex conjugate vector space as follows:
 is also a representation, as one may check explicitly.
If is a real Lie algebra and is a representation of it over the vector space , then the conjugate representation is defined over the conjugate vector space as follows:
 is also a representation, as one may check explicitly.
If two real Lie algebras have the same complexification, and we have a complex representation of the complexified Lie algebra, their conjugate representations are still going to be different. See spinor for some examples associated with spinor representations of the spin groups and .
If formula_1 is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket),
For a finite-dimensional unitary representation, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations.


Andrei Suslin

Andrei Suslin (, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee Chair and Professor of mathematics at Northwestern University.
He was born on 27 December 1950 in St. Petersburg, Russia. As a young person, he was an "all Leningrad" gymnast. He received his PhD from Leningrad University in 1974; his thesis was titled "Projective modules over polynomial rings".
In 1976 he and Daniel Quillen independently proved Serre's conjecture about the triviality of algebraic vector bundles on affine space.
In 1982 he and Alexander Merkurjev proved the Merkurjev–Suslin theorem on the norm residue homomorphism in Milnor K2-theory, with applications to the Brauer group.
Suslin was an invited speaker at the International Congress of Mathematicians in 1978 and 1994, and he gave a plenary invited address at the Congress in 1986. He was awarded the Frank Nelson Cole Prize in Algebra in 2000 by the American Mathematical Society for his work on motivic cohomology.
In 2010 special issues of "Journal of K-theory"
and of "Documenta Mathematica"
 have been published in honour of his 60th birthday.
He died on 10 July 2018.


Self-adjoint

In mathematics, an element "x" of a *-algebra is self-adjoint if formula_1.
A collection "C" of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if formula_2 then since formula_3 in a star-algebra, the set {"x","y"} is a self-adjoint set even though "x" and "y" need not be self-adjoint elements.
In functional analysis, a linear operator "A" on a Hilbert space is called self-adjoint if it is equal to its own adjoint "A" and that the domain of "A" is the same as that of "A". See self-adjoint operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator "A" is self-adjoint if and only if the matrix describing "A" with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. Hermitian matrices are also called self-adjoint.
In a dagger category, a morphism formula_4 is called self-adjoint if formula_5; this is possible only for an endomorphism formula_6.


Free category

In mathematics, the free category or path category generated by a directed graph or quiver is the category that results from freely concatenating arrows together, whenever the target of one arrow is the source of the next.
More precisely, the objects of the category are the vertices of the quiver, and the morphisms are paths between objects. Here, a path is defined as a finite sequence
where formula_2 is a vertex of the quiver, formula_3 is an edge of the quiver, and "n" ranges over the non-negative integers. For every vertex formula_4 of the quiver, there is an "empty path" which constitutes the identity morphisms of the category.
The composition operation is concatenation of paths. Given paths
their composition is
Note that the result of the composition starts with the right operand of the composition, and ends with its left operand.
Properties.
The category of small categories Cat has a forgetful functor into the quiver category Quiv:
which takes objects to vertices and morphisms to arrows. Intuitively, "[forgets] which arrows are composites and which are identities". This forgetful functor is right adjoint to the functor sending a quiver to the corresponding free category.
Universal property.
The free category on a quiver can be described up to isomorphism by a universal property. Let : Quiv → Cat be the functor that takes a quiver to the free category on that quiver (as described above), let be the forgetful functor defined above, and let be any quiver. Then there is a graph homomorphism : → (()) and given any category D and any graph homomorphism : → , there is a unique functor : () → D such that ()∘=, i.e. the following diagram commutes:
The functor is left adjoint to the forgetful functor .


Centrally-closed subgroup

In mathematics, in the realm of group theory, a subgroup of a group is said to be centrally closed if the centralizer of any nonidentity element of the subgroup lies inside the subgroup.
Some facts about centrally closed subgroups:


Simple Lie group

In mathematics, a simple Lie group is a connected non-abelian Lie group "G" which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces.
Together with the commutative Lie group of the real numbers, formula_1, and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL("n") of "n" by "n" matrices with determinant equal to 1 is simple for all "n" > 1.
The simple Lie groups were first classified by Wilhelm Killing and later perfected by Élie Cartan. This classification is often referred to as Killing-Cartan classification.
Definition.
Unfortunately, there is no universally accepted definition of a simple Lie group. In particular, it is not always defined as a Lie group that is simple as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether is a simple Lie group.
The most common definition is that a Lie group is simple if it is connected, non-abelian, and every closed "connected" normal subgroup is either the identity or the whole group. In particular, simple groups are allowed to have a non-trivial center, but is not simple.
In this article the connected simple Lie groups with trivial center are listed. Once these are known, the ones with non-trivial center are easy to list as follows. Any simple Lie group with trivial center has a universal cover, whose center is the fundamental group of the simple Lie group. The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the center.
Alternatives.
An equivalent definition of a simple Lie group follows from the Lie correspondence: A connected Lie group is simple if its Lie algebra is simple. An important technical point is that a simple Lie group may contain "discrete" normal subgroups, hence being a simple Lie group is different from being simple as an abstract group.
Simple Lie groups include many classical Lie groups, which provide a group-theoretic underpinning for spherical geometry, projective geometry and related geometries in the sense of Felix Klein's Erlangen program. It emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry. These "exceptional groups" account for many special examples and configurations in other branches of mathematics, as well as contemporary theoretical physics.
As a counterexample, the general linear group is neither simple, nor semisimple. This is because multiples of the identity form a nontrivial normal subgroup, thus evading the definition. Equivalently, the corresponding Lie algebra has a degenerate Killing form, because multiples of the identity map to the zero element of the algebra. Thus, the corresponding Lie algebra is also neither simple nor semisimple. Another counter-example are the special orthogonal groups in even dimension. These have the matrix formula_2 in the center, and this element is path-connected to the identity element, and so these groups evade the definition. Both of these are reductive groups.
Related ideas.
Simple Lie algebras.
The Lie algebra of a simple Lie group is a simple Lie algebra. This is a one-to-one correspondence between connected simple Lie groups with trivial center and simple Lie algebras of dimension greater than 1. (Authors differ on whether the one-dimensional Lie algebra should be counted as simple.)
Over the complex numbers the semisimple Lie algebras are classified by their Dynkin diagrams, of types "ABCDEFG". If "L" is a real simple Lie algebra, its complexification is a simple complex Lie algebra, unless "L" is already the complexification of a Lie algebra, in which case the complexification of "L" is a product of two copies of "L". This reduces the problem of classifying the real simple Lie algebras to that of finding all the real forms of each complex simple Lie algebra (i.e., real Lie algebras whose complexification is the given complex Lie algebra). There are always at least 2 such forms: a split form and a compact form, and there are usually a few others. The different real forms correspond to the classes of automorphisms of order at most 2 of the complex Lie algebra.
Symmetric spaces.
Symmetric spaces are classified as follows.
First, the universal cover of a symmetric space is still symmetric, so we can reduce to the case of simply connected symmetric spaces. (For example, the universal cover of a real projective plane is a sphere.)
Second, the product of symmetric spaces is symmetric, so we may as well just classify the irreducible simply connected ones (where irreducible means they cannot be written as a product of smaller symmetric spaces).
The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to each "non-compact" simple Lie group "G",
one compact and one non-compact. The non-compact one is a cover of the quotient of "G" by a maximal compact subgroup "H", and the compact one is a cover of the quotient of
the compact form of "G" by the same subgroup "H". This duality between compact and non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry.
Hermitian symmetric spaces.
A symmetric space with a compatible complex structure is called Hermitian. The compact simply connected irreducible Hermitian symmetric spaces fall into 4 infinite families with 2 exceptional ones left over, and each has a non-compact dual. In addition the complex plane is also a Hermitian symmetric space; this gives the complete list of irreducible Hermitian symmetric spaces.
The four families are the types A III, B I and D I for , D III, and C I, and the two exceptional ones are types E III and E VII of complex dimensions 16 and 27.
Notation.
formula_3  stand for the real numbers, complex numbers, quaternions, and octonions.
In the symbols such as "E"6−26 for the exceptional groups, the exponent −26 is the signature of an invariant symmetric bilinear form that is negative definite on the maximal compact subgroup. It is equal to the dimension of the group minus twice the dimension of a maximal compact subgroup.
The fundamental group listed in the table below is the fundamental group of the simple group with trivial center.
Other simple groups with the same Lie algebra correspond to subgroups of this fundamental group (modulo the action of the outer automorphism group).
Full classification.
Simple Lie groups are fully classified. The classification is usually stated in several steps, namely:
One can show that the fundamental group of any Lie group is a discrete commutative group. Given a (nontrivial) subgroup formula_7 of the fundamental group of some Lie group formula_5, one can use the theory of covering spaces to construct a new group formula_9 with formula_10 in its center. Now any (real or complex) Lie group can be obtained by applying this construction to centerless Lie groups. Note that real Lie groups obtained this way might not be real forms of any complex group. A very important example of such a real group is the metaplectic group, which appears in infinite-dimensional representation theory and physics. When one takes for formula_7 the full fundamental group, the resulting Lie group formula_12 is the universal cover of the centerless Lie group formula_5, and is simply connected. In particular, every (real or complex) Lie algebra also corresponds to a unique connected and simply connected Lie group formula_14 with that Lie algebra, called the "simply connected Lie group" associated to formula_15
Compact Lie groups.
Every simple complex Lie algebra has a unique real form whose corresponding centerless Lie group is compact. It turns out that the simply connected Lie group in these cases is also compact. Compact Lie groups have a particularly tractable representation theory because of the Peter–Weyl theorem. Just like simple complex Lie algebras, centerless compact Lie groups are classified by Dynkin diagrams (first classified by Wilhelm Killing and Élie Cartan).
For the infinite (A, B, C, D) series of Dynkin diagrams, the simply connected compact Lie group associated to each Dynkin diagram can be explicitly described as a matrix group, with the corresponding centerless compact Lie group described as the quotient by a subgroup of scalar matrices.
Overview of the classification.
A"r" has as its associated simply connected compact group the special unitary group, SU("r" + 1) and as its associated centerless compact group the projective unitary group PU("r" + 1).
B"r" has as its associated centerless compact groups the odd special orthogonal groups, SO(2"r" + 1). This group is not simply connected however: its universal (double) cover is the Spin group.
C"r" has as its associated simply connected group the group of unitary symplectic matrices, Sp("r") and as its associated centerless group the Lie group PSp("r") = Sp("r")/{I, −I} of projective unitary symplectic matrices. The symplectic groups have a double-cover by the metaplectic group.
D"r" has as its associated compact group the even special orthogonal groups, SO(2"r") and as its associated centerless compact group the projective special orthogonal group PSO(2"r") = SO(2"r")/{I, −I}. As with the B series, SO(2"r") is not simply connected; its universal cover is again the spin group, but the latter again has a center (cf. its article).
The diagram D2 is two isolated nodes, the same as A1 ∪ A1, and this coincidence corresponds to the covering map homomorphism from SU(2) × SU(2) to SO(4) given by quaternion multiplication; see quaternions and spatial rotation. Thus SO(4) is not a simple group. Also, the diagram D3 is the same as A3, corresponding to a covering map homomorphism from SU(4) to SO(6).
In addition to the four families "A""i", "B""i", "C""i", and "D""i" above, there are five so-called exceptional Dynkin diagrams G2, F4, E6, E7, and E8; these exceptional Dynkin diagrams also have associated simply connected and centerless compact groups. However, the groups associated to the exceptional families are more difficult to describe than those associated to the infinite families, largely because their descriptions make use of exceptional objects. For example, the group associated to G2 is the automorphism group of the octonions, and the group associated to F4 is the automorphism group of a certain Albert algebra.
See also E7½.
Simple Lie groups of small dimension.
The following table lists some Lie groups with simple Lie algebras of small
dimension. The groups on a given line all have the same Lie algebra. In the dimension 1 case, the groups are abelian and not simple.
Simply laced groups.
A simply laced group is a Lie group whose Dynkin diagram only contain simple links, and therefore all the nonzero roots of the corresponding Lie algebra have the same length. The A, D and E series groups are all simply laced, but no group of type B, C, F, or G is simply laced.


Algebraic differential equation

In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used.
The intention is to include equations formed by means of differential operators, in which the coefficients are rational functions of the variables (e.g. the hypergeometric equation). Algebraic differential equations are widely used in computer algebra and number theory.
A simple concept is that of a polynomial vector field, in other words a vector field expressed with respect to a standard co-ordinate basis as the first partial derivatives with polynomial coefficients. This is a type of first-order algebraic differential operator.
Algebraic solutions.
It is usually not the case that the general solution of an algebraic differential equation is an algebraic function: solving equations typically produces novel transcendental functions. The case of algebraic solutions is however of considerable interest; the classical Schwarz list deals with the case of the hypergeometric equation. In differential Galois theory the case of algebraic solutions is that in which the differential Galois group "G" is finite (equivalently, of dimension 0, or of a finite monodromy group for the case of Riemann surfaces and linear equations). This case stands in relation with the whole theory roughly as invariant theory does to group representation theory. The group "G" is in general difficult to compute, the understanding of algebraic solutions is an indication of upper bounds for "G".


Polynomial function


Lattice (module)

In mathematics, in the field of ring theory, a lattice is a module over a ring which is embedded in a vector space over a field, giving an algebraic generalisation of the way a lattice group is embedded in a real vector space.
Formal definition.
Let "R" be an integral domain with field of fractions "K". An "R"-submodule "M" of a "K"-vector space "V" is a "lattice" if "M" is finitely generated over "R". It is "full" if "V" = "K"·"M".
Pure sublattices.
An "R"-submodule "N" of "M" that is itself a lattice is an "R"-pure sublattice if "M"/"N" is "R"-torsion-free. There is a one-to-one correspondence between "R"-pure sublattices "N" of "M" and "K"-subspaces "W" of "V", given by


Analytic geometry

In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.
History.
Ancient Greece.
The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry.
Apollonius of Perga, in "On Determinate Section", dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the "Conics" further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve "a posteriori" instead of "a priori". That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.
Persia.
The 11th-century Persian mathematician Omar Khayyam saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical and geometric algebra with his geometric solution of the general cubic equations, but the decisive step came later with Descartes. Omar Khayyam is credited with identifying the foundations of algebraic geometry, and his book "Treatise on Demonstrations of Problems of Algebra" (1070), which laid down the principles of analytic geometry, is part of the body of Persian mathematics that was eventually transmitted to Europe. Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry.
Western Europe.
Analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. "Cartesian geometry", the alternative term used for analytic geometry, is named after Descartes.
Descartes made significant progress with the methods in an essay titled "La Geometrie (Geometry)", one of the three accompanying essays (appendices) published in 1637 together with his "Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences", commonly referred to as "Discourse on Method".
"La Geometrie", written in his native French tongue, and its philosophical principles, provided a foundation for calculus in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition.
Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of "Ad locos planos et solidos isagoge" (Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes' "Discourse". Clearly written and well received, the "Introduction" also laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described the geometric curve that satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves. As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonhard Euler who first applied the coordinate method in a systematic study of space curves and surfaces.
Coordinates.
In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. Similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but the most common are the following:
Cartesian coordinates (in a plane or space).
The most common coordinate system to use is the Cartesian coordinate system, where each point has an "x"-coordinate representing its horizontal position, and a "y"-coordinate representing its vertical position. These are typically written as an ordered pair ("x", "y"). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates ("x", "y", "z").
Polar coordinates (in a plane).
In polar coordinates, every point of the plane is represented by its distance "r" from the origin and its angle "θ", with "θ" normally measured counterclockwise from the positive "x"-axis. Using this notation, points are typically written as an ordered pair ("r", "θ"). One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae: formula_1 . This system may be generalized to three-dimensional space through the use of cylindrical or spherical coordinates.
Cylindrical coordinates (in a space).
In cylindrical coordinates, every point of space is represented by its height "z", its radius "r" from the "z"-axis and the angle "θ" its projection on the "xy"-plane makes with respect to the horizontal axis.
Spherical coordinates (in a space).
In spherical coordinates, every point in space is represented by its distance "ρ" from the origin, the angle "θ" its projection on the "xy"-plane makes with respect to the horizontal axis, and the angle "φ" that it makes with respect to the "z"-axis. The names of the angles are often reversed in physics.
Equations and curves.
In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation, or locus. For example, the equation "y" = "x" corresponds to the set of all the points on the plane whose "x"-coordinate and "y"-coordinate are equal. These points form a line, and "y" = "x" is said to be the equation for this line. In general, linear equations involving "x" and "y" specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.
Usually, a single equation corresponds to a curve on the plane. This is not always the case: the trivial equation "x" = "x" specifies the entire plane, and the equation "x"2 + "y"2 = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces (see below), or as a system of parametric equations. The equation "x"2 + "y"2 = "r"2 is the equation for any circle centered at the origin (0, 0) with a radius of r.
Lines and planes.
Lines in a Cartesian plane, or more generally, in affine coordinates, can be described algebraically by "linear" equations. In two dimensions, the equation for non-vertical lines is often given in the "slope-intercept form":
where:
In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".
Specifically, let formula_3 be the position vector of some point formula_4, and let formula_5 be a nonzero vector. The plane determined by this point and vector consists of those points formula_6, with position vector formula_7, such that the vector drawn from formula_8 to formula_6 is perpendicular to formula_10. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points formula_7 such that
Expanded this becomes
which is the "point-normal" form of the equation of a plane. This is just a linear equation:
Conversely, it is easily shown that if "a", "b", "c" and "d" are constants and "a", "b", and "c" are not all zero, then the graph of the equation
is a plane having the vector formula_5 as a normal. This familiar equation for a plane is called the "general form" of the equation of the plane.
In three dimensions, lines can "not" be described by a single linear equation, so they are frequently described by parametric equations:
where:
Conic sections.
In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form
As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional projective space formula_21
The conic sections described by this equation can be classified using the discriminant
If the conic is non-degenerate, then:
Quadric surfaces.
A quadric, or quadric surface, is a "2"-dimensional surface in 3-dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates , the general quadric is defined by the algebraic equation
Quadric surfaces include ellipsoids (including the sphere), paraboloids, hyperboloids, cylinders, cones, and planes.
Distance and angle.
In analytic geometry, geometric notions such as distance and angle measure are defined using formulas. These definitions are designed to be consistent with the underlying Euclidean geometry. For example, using Cartesian coordinates on the plane, the distance between two points ("x"1, "y"1) and ("x"2, "y"2) is defined by the formula
which can be viewed as a version of the Pythagorean theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula
where "m" is the slope of the line.
In three dimensions, distance is given by the generalization of the Pythagorean theorem:
while the angle between two vectors is given by the dot product. The dot product of two Euclidean vectors A and B is defined by
where θ is the angle between A and B.
Transformations.
Transformations are applied to a parent function to turn it into a new function with similar characteristics.
The graph of formula_34 is changed by standard transformations as follows:
There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered.
For more information, consult the Wikipedia article on affine transformations.
For example, the parent function formula_52 has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant. In general, if formula_53, then it can be transformed into formula_54. In the new transformed function, formula_55 is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative formula_55 values, the function is reflected in the formula_35-axis. The formula_43 value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like formula_55, reflects the function in the formula_38-axis when it is negative. The formula_40 and formula_37 values introduce translations, formula_37, vertical, and formula_40 horizontal. Positive formula_37 and formula_40 values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.
Transformations can be applied to any geometric equation whether or not the equation represents a function.
Transformations can be considered as individual transactions or in combinations.
Suppose that formula_34 is a relation in the formula_68 plane. For example,
is the relation that describes the unit circle.
Finding intersections of geometric objects.
For two geometric objects P and Q represented by the relations formula_70 and formula_71 the intersection is the collection of all points formula_72 which are in both relations.
For example, formula_6 might be the circle with radius 1 and center formula_74: formula_75 and formula_76 might be the circle with radius 1 and center formula_77. The intersection of these two circles is the collection of points which make both equations true. Does the point formula_74 make both equations true? Using formula_74 for formula_72, the equation for formula_76 becomes formula_82 or formula_83 which is true, so formula_74 is in the relation formula_76. On the other hand, still using formula_74 for formula_72 the equation for formula_6 becomes formula_89 or formula_90 which is false. formula_74 is not in formula_6 so it is not in the intersection.
The intersection of formula_6 and formula_76 can be found by solving the simultaneous equations:
Traditional methods for finding intersections include substitution and elimination.
Substitution: Solve the first equation for formula_38 in terms of formula_35 and then substitute the expression for formula_38 into the second equation:
We then substitute this value for formula_102 into the other equation and proceed to solve for formula_35:
Next, we place this value of formula_35 in either of the original equations and solve for formula_38:
So our intersection has two points:
Elimination: Add (or subtract) a multiple of one equation to the other equation so that one of the variables is eliminated. For our current example, if we subtract the first equation from the second we get formula_114. The formula_102 in the first equation is subtracted from the formula_102 in the second equation leaving no formula_38 term. The variable formula_38 has been eliminated. We then solve the remaining equation for formula_35, in the same way as in the substitution method:
We then place this value of formula_35 in either of the original equations and solve for formula_38:
So our intersection has two points:
For conic sections, as many as 4 points might be in the intersection.
Finding intercepts.
One type of intersection which is widely studied is the intersection of a geometric object with the formula_35 and formula_38 coordinate axes.
The intersection of a geometric object and the formula_38-axis is called the formula_38-intercept of the object.
The intersection of a geometric object and the formula_35-axis is called the formula_35-intercept of the object.
For the line formula_135, the parameter formula_43 specifies the point where the line crosses the formula_38 axis. Depending on the context, either formula_43 or the point formula_139 is called the formula_38-intercept.
Tangents and normals.
Tangent lines and planes.
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Informally, it is a line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where "f" is the derivative of "f". A similar definition applies to space curves and curves in "n"-dimensional Euclidean space.
As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.
Normal line and vector.
In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.
In the three-dimensional case a surface normal, or simply normal, to a surface at a point "P" is a vector that is perpendicular to the tangent plane to that surface at "P". The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality.


Character (mathematics)

In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified.
Multiplicative character.
A multiplicative character (or linear character, or simply character) on a group "G" is a group homomorphism from "G" to the multiplicative group of a field , usually the field of complex numbers. If "G" is any group, then the set Ch("G") of these morphisms forms an abelian group under pointwise multiplication.
This group is referred to as the character group of "G". Sometimes only "unitary" characters are considered (thus the image is in the unit circle); other such homomorphisms are then called "quasi-characters". Dirichlet characters can be seen as a special case of this definition.
Multiplicative characters are linearly independent, i.e. if formula_1 are different characters on a group "G" then from formula_2 it follows that formula_3.
Character of a representation.
The character formula_4 of a representation formula_5 of a group "G" on a finite-dimensional vector space "V" over a field "F" is the trace of the representation formula_6 , i.e.
In general, the trace is not a group homomorphism, nor does the set of traces form a group. The characters of one-dimensional representations are identical to one-dimensional representations, so the above notion of multiplicative character can be seen as a special case of higher-dimensional characters. The study of representations using characters is called "character theory" and one-dimensional characters are also called "linear characters" within this context.
Alternative definition.
If restricted to finite abelian group with formula_9 representation in formula_10 (i.e. formula_11), the following alternative definition would be equivalent to the above (For abelian groups, every matrix representation decomposes into a direct sum of formula_9 representations. For non-abelian groups, the original definition would be more general than this one):
A character formula_13 of group formula_14 is a mapping formula_15 such that formula_16 for all formula_17
If formula_18 is a finite abelian group, the characters play the role of harmonics. For infinite abelian groups, the above would be replaced by formula_19 where formula_20 is the circle group.


Heisuke Hironaka

 is a Japanese mathematician who was awarded the Fields Medal in 1970 for his contributions to algebraic geometry.
Career.
Hironaka entered Kyoto University in 1949. After completing his undergraduate studies at Kyoto University, he received his Ph.D. in 1960 from Harvard while under the direction of Oscar Zariski.
Hironaka held teaching positions at Brandeis University from 1960-1963, Columbia University in 1964, and Kyoto University from 1975 to 1988. He was a professor of mathematics at Harvard University from 1968 until becoming "emeritus" in 1992 and was a president of Yamaguchi University from 1996 to 2002.
Research.
In 1964, Hironaka proved that singularities of algebraic varieties admit resolutions in characteristic zero. This means that any algebraic variety can be replaced by (more precisely is birationally equivalent to) a similar variety which has no singularities. He also introduced Hironaka's example showing that a deformation of Kähler manifolds need not be Kähler. In 2017 he posted to his personal webpage a manuscript that claims to prove the existence of a resolution of singularities in positive characteristic.
Awards.
Hironaka was awarded the Fields Medal in 1970.
Personal life.
Hironaka has been active in raising funds for causes such as mathematical education. His wife Wakako Hironaka is a politician. His daughter, Eriko Hironaka, is also a mathematician and focuses on low-dimensional topology and geometric topology.


Isoclinism of groups

In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by to help classify and understand p-groups, although it is applicable to all groups. Isoclinism also has consequences for the Schur multiplier and the associated aspects of character theory, as described in and . The word "isoclinism" comes from the Greek ισοκλινης meaning equal slope.
Some textbooks discussing isoclinism include and and .
Definition.
The isoclinism class of a group "G" is determined by the groups "G"/"Z"("G") (the inner automorphism group) and "G"′ (the commutator subgroup) and the commutator map from "G"/"Z"("G") × "G"/"Z"("G") to "G"′ (taking "a", "b" to "aba−1b−1").
In other words, two groups "G"1 and "G"2 are isoclinic if there are isomorphisms from "G"1/"Z"("G"1) to "G"2/"Z"("G"2) and from "G"1′ to "G"2′ commuting with the commutator map.
Examples.
All Abelian groups are isoclinic since they are equal to their centers and their commutator subgroups are always the identity subgroup. Indeed, a group is isoclinic to an abelian group if and only if it is itself abelian, and "G" is isoclinic with "G"×"A" if and only if "A" is abelian. The dihedral, quasidihedral, and quaternion groups of order 2"n" are isoclinic for "n"≥3, in more detail.
Isoclinism divides "p"-groups into families, and the smallest members of each family are called stem groups. A group is a stem group if and only if Z("G") ≤ ["G","G"], that is, if and only if every element of the center of the group is contained in the derived subgroup (also called the commutator subgroup), . Some enumeration results on isoclinism families are given in .
Isoclinism is used in theory of projective representations of finite groups, as all Schur covering groups of a group are isoclinic, a fact already hinted at by Hall according to . This is used in describing the character tables of the finite simple groups .


Classical group

In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph "The Classical Groups".
The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern physics. A few examples are the following. The rotation group is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group is a symmetry group of spacetime of special relativity. The special unitary group is the symmetry group of quantum chromodynamics and the symplectic group finds application in Hamiltonian mechanics and quantum mechanical versions of it.
The classical groups.
The classical groups are exactly the general linear groups over and together with the automorphism groups of non-degenerate forms discussed below. These groups are usually additionally restricted to the subgroups whose elements have determinant 1, so that their centers are discrete. The classical groups, with the determinant 1 condition, are listed in the table below. In the sequel, the determinant 1 condition is "not" used consistently in the interest of greater generality.
The complex classical groups are , and . A group is complex according to whether its Lie algebra is complex. The real classical groups refers to all of the classical groups since any Lie algebra is a real algebra. The compact classical groups are the compact real forms of the complex classical groups. These are, in turn, , and . One characterization of the compact real form is in terms of the Lie algebra . If , the complexification of , and if the connected group generated by } is compact, then is a compact real form.
The classical groups can uniformly be characterized in a different way using real forms. The classical groups (here with the determinant 1 condition, but this is not necessary) are the following:
For instance, is a real form of , is a real form of , and is a real form of . Without the determinant 1 condition, replace the special linear groups with the corresponding general linear groups in the characterization. The algebraic groups in question are Lie groups, but the "algebraic" qualifier is needed to get the right notion of "real form".
Bilinear and sesquilinear forms.
The classical groups are defined in terms of forms defined on , , and , where and are the fields of the real and complex numbers. The quaternions, , do not constitute a field because multiplication does not commute; they form a division ring or a skew field or non-commutative field. However, it is still possible to define matrix quaternionic groups. For this reason, a vector space is allowed to be defined over , , as well as below. In the case of , is a "right" vector space to make possible the representation of the group action as matrix multiplication from the "left", just as for and .
A form on some finite-dimensional right vector space over , or is bilinear if
It is called sesquilinear if
These conventions are chosen because they work in all cases considered. An automorphism of is a map in the set of linear operators on such that
The set of all automorphisms of form a group, it is called the automorphism group of , denoted . This leads to a preliminary definition of a classical group:
This definition has some redundancy. In the case of , bilinear is equivalent to sesquilinear. In the case of , there are no non-zero bilinear forms.
Symmetric, skew-symmetric, Hermitian, and skew-Hermitian forms.
A form is symmetric if
It is skew-symmetric if
It is Hermitian if
Finally, it is skew-Hermitian if
A bilinear form is uniquely a sum of a symmetric form and a skew-symmetric form. A transformation preserving preserves both parts separately. The groups preserving symmetric and skew-symmetric forms can thus be studied separately. The same applies, mutatis mutandis, to Hermitian and skew-Hermitian forms. For this reason, for the purposes of classification, only purely symmetric, skew-symmetric, Hermitian, or skew-Hermitian forms are considered. The normal forms of the forms correspond to specific suitable choices of bases. These are bases giving the following normal forms in coordinates:
The in the skew-Hermitian form is the third basis element in the basis for . Proof of existence of these bases and Sylvester's law of inertia, the independence of the number of plus- and minus-signs, and , in the symmetric and Hermitian forms, as well as the presence or absence of the fields in each expression, can be found in or . The pair , and sometimes , is called the signature of the form.
Explanation of occurrence of the fields : There are no nontrivial bilinear forms over . In the symmetric bilinear case, only forms over have a signature. In other words, a complex bilinear form with "signature" can, by a change of basis, be reduced to a form where all signs are "" in the above expression, whereas this is impossible in the real case, in which is independent of the basis when put into this form. However, Hermitian forms have basis-independent signature in both the complex and the quaternionic case. (The real case reduces to the symmetric case.) A skew-Hermitian form on a complex vector space is rendered Hermitian by multiplication by , so in this case, only is interesting.
Automorphism groups.
The first section presents the general framework. The other sections exhaust the qualitatively different cases that arise as automorphism groups of bilinear and sesquilinear forms on finite-dimensional vector spaces over , and .
Aut("φ") – the automorphism group.
Assume that is a non-degenerate form on a finite-dimensional vector space over or . The automorphism group is defined, based on condition (), as
Every has an adjoint with respect to defined by
Using this definition in condition (), the automorphism group is seen to be given by
Fix a basis for . In terms of this basis, put
where are the components of . This is appropriate for the bilinear forms. Sesquilinear forms have similar expressions and are treated separately later. In matrix notation one finds
and
from () where is the matrix . The non-degeneracy condition means precisely that is invertible, so the adjoint always exists. expressed with this becomes
The Lie algebra of the automorphism groups can be written down immediately. Abstractly, if and only if
for all , corresponding to the condition in () under the exponential mapping of Lie algebras, so that
or in a basis
as is seen using the power series expansion of the exponential mapping and the linearity of the involved operations. Conversely, suppose that . Then, using the above result, . Thus the Lie algebra can be characterized without reference to a basis, or the adjoint, as
The normal form for will be given for each classical group below. From that normal form, the matrix can be read off directly. Consequently, expressions for the adjoint and the Lie algebras can be obtained using formulas () and (). This is demonstrated below in most of the non-trivial cases.
Bilinear case.
When the form is symmetric, is called . When it is skew-symmetric then is called . This applies to the real and the complex cases. The quaternionic case is empty since no nonzero bilinear forms exists on quaternionic vector spaces.
Real case.
The real case breaks up into two cases, the symmetric and the antisymmetric forms that should be treated separately.
O("p", "q") and O("n") – the orthogonal groups.
If is symmetric and the vector space is real, a basis may be chosen so that
The number of plus and minus-signs is independent of the particular basis. In the case one writes where is the number of plus signs and is the number of minus-signs, . If the notation is . The matrix is in this case
after reordering the basis if necessary. The adjoint operation () then becomes
which reduces to the usual transpose when or is 0. The Lie algebra is found using equation () and a suitable ansatz (this is detailed for the case of below),
and the group according to () is given by
The groups and are isomorphic through the map
For example, the Lie algebra of the Lorentz group could be written as
Naturally, it is possible to rearrange so that the -block is the upper left (or any other block). Here the "time component" end up as the fourth coordinate in a physical interpretation, and not the first as may be more common.
Sp("m", R) – the real symplectic group.
If is skew-symmetric and the vector space is real, there is a basis giving
where . For one writes In case one writes or . From the normal form one reads off
By making the ansatz
where are -dimensional matrices and considering (),
one finds the Lie algebra of ,
and the group is given by
Complex case.
Like in the real case, there are two cases, the symmetric and the antisymmetric case that each yield a family of classical groups.
O("n", C) – the complex orthogonal group.
If case is symmetric and the vector space is complex, a basis
with only plus-signs can be used. The automorphism group is in the case of called . The lie algebra is simply a special case of that for ,
and the group is given by
In terms of classification of simple Lie algebras, the are split into two classes, those with odd with root system and even with root system .
Sp("m", C) – the complex symplectic group.
For skew-symmetric and the vector space complex, the same formula,
applies as in the real case. For one writes In case one writes or . The Lie algebra parallels that of ,
and the group is given by
Sesquilinear case.
In the sequilinear case, one makes a slightly different approach for the form in terms of a basis,
The other expressions that get modified are
The real case, of course, provides nothing new. The complex and the quaternionic case will be considered below.
Complex case.
From a qualitative point of view, consideration of skew-Hermitian forms (up to isomorphism) provide no new groups; multiplication by renders a skew-Hermitian form Hermitian, and vice versa. Thus only the Hermitian case needs to be considered.
U("p", "q") and U("n") – the unitary groups.
A non-degenerate hermitian form has the normal form
As in the bilinear case, the signature ("p", "q") is independent of the basis. The automorphism group is denoted , or, in the case of , . If the notation is . In this case, takes the form
and the Lie algebra is given by
The group is given by
As a comparison, a Unitary matrix U(n) is defined as
formula_45
We note that formula_46 is the same as formula_47
Quaternionic case.
The space is considered as a "right" vector space over . This way, for a quaternion , a quaternion column vector and quaternion matrix . If was a "left" vector space over , then matrix multiplication from the "right" on row vectors would be required to maintain linearity. This does not correspond to the usual linear operation of a group on a vector space when a basis is given, which is matrix multiplication from the "left" on column vectors. Thus is henceforth a right vector space over . Even so, care must be taken due to the non-commutative nature of . The (mostly obvious) details are skipped because complex representations will be used.
When dealing with quaternionic groups it is convenient to represent quaternions using complex ,
With this representation, quaternionic multiplication becomes matrix multiplication and quaternionic conjugation becomes taking the Hermitian adjoint. Moreover, if a quaternion according to the complex encoding is given as a column vector , then multiplication from the left by a matrix representation of a quaternion produces a new column vector representing the correct quaternion. This representation differs slightly from a more common representation found in the quaternion article. The more common convention would force multiplication from the right on a row matrix to achieve the same thing.
Incidentally, the representation above makes it clear that the group of unit quaternions () is isomorphic to .
Quaternionic -matrices can, by obvious extension, be represented by block-matrices of complex numbers. If one agrees to represent a quaternionic column vector by a column vector with complex numbers according to the encoding of above, with the upper numbers being the and the lower the , then a quaternionic -matrix becomes a complex -matrix exactly of the form given above, but now with α and β -matrices. More formally
A matrix has the form displayed in () if and only if . With these identifications,
The space is a real algebra, but it is not a complex subspace of . Multiplication (from the left) by in using entry-wise quaternionic multiplication and then mapping to the image in yields a different result than multiplying entry-wise by directly in . The quaternionic multiplication rules give where the new and are inside the parentheses.
The action of the quaternionic matrices on quaternionic vectors is now represented by complex quantities, but otherwise it is the same as for "ordinary" matrices and vectors. The quaternionic groups are thus embedded in where is the dimension of the quaternionic matrices.
The determinant of a quaternionic matrix is defined in this representation as being the ordinary complex determinant of its representative matrix. The non-commutative nature of quaternionic multiplication would, in the quaternionic representation of matrices, be ambiguous. The way is embedded in is not unique, but all such embeddings are related through for , leaving the determinant unaffected. The name of in this complex guise is .
As opposed to in the case of , both the Hermitian and the skew-Hermitian case bring in something new when is considered, so these cases are considered separately.
GL("n", H) and SL("n", H).
Under the identification above,
Its Lie algebra is the set of all matrices in the image of the mapping of above,
The quaternionic special linear group is given by
where the determinant is taken on the matrices in . Alternatively, one can define this as the kernel of the Dieudonné determinant formula_52. The Lie algebra is
Sp("p", "q") – the quaternionic unitary group.
As above in the complex case, the normal form is
and the number of plus-signs is independent of basis. When with this form, . The reason for the notation is that the group can be represented, using the above prescription, as a subgroup of preserving a complex-hermitian form of signature If or the group is denoted . It is sometimes called the hyperunitary group.
In quaternionic notation,
meaning that "quaternionic" matrices of the form
will satisfy
see the section about . Caution needs to be exercised when dealing with quaternionic matrix multiplication, but here only and are involved and these commute with every quaternion matrix. Now apply prescription () to each block,
and the relations in () will be satisfied if
The Lie algebra becomes
The group is given by
Returning to the normal form of for , make the substitutions and with . Then
viewed as a -valued form on . Thus the elements of , viewed as linear transformations of , preserve both a Hermitian form of signature and a non-degenerate skew-symmetric form. Both forms take purely complex values and due to the prefactor of of the second form, they are separately conserved. This means that
and this explains both the name of the group and the notation.
O∗(2"n")= O("n", H)- quaternionic orthogonal group.
The normal form for a skew-hermitian form is given by
where is the third basis quaternion in the ordered listing . In this case, may be realized, using the complex matrix encoding of above, as a subgroup of which preserves a non-degenerate complex skew-hermitian form of signature . From the normal form one sees that in quaternionic notation
and from () follows that
for . Now put
according to prescription (). The same prescription yields for ,
Now the last condition in () in complex notation reads
The Lie algebra becomes
and the group is given by
The group can be characterized as
where the map is defined by .
Also, the form determining the group can be viewed as a -valued form on . Make the substitutions and in the expression for the form. Then
The form is Hermitian (while the first form on the left hand side is skew-Hermitian) of signature . The signature is made evident by a change of basis from to where are the first and last basis vectors respectively. The second form, is symmetric positive definite. Thus, due to the factor , preserves both separately and it may be concluded that
and the notation "O" is explained.
Classical groups over general fields or algebras.
Classical groups, more broadly considered in algebra, provide particularly interesting matrix groups. When the field "F" of coefficients of the matrix group is either real number or complex numbers, these groups are just the classical Lie groups. When the ground field is a finite field, then the classical groups are groups of Lie type. These groups play an important role in the classification of finite simple groups. Also, one may consider classical groups over a unital associative algebra "R" over "F"; where "R" = H (an algebra over reals) represents an important case. For the sake of generality the article will refer to groups over "R", where "R" may be the ground field "F" itself.
Considering their abstract group theory, many linear groups have a "special" subgroup, usually consisting of the elements of determinant 1 over the ground field, and most of them have associated "projective" quotients, which are the quotients by the center of the group. For orthogonal groups in characteristic 2 "S" has a different meaning.
The word "general" in front of a group name usually means that the group is allowed to multiply some sort of form by a constant, rather than leaving it fixed. The subscript "n" usually indicates the dimension of the module on which the group is acting; it is a vector space if "R" = "F". Caveat: this notation clashes somewhat with the "n" of Dynkin diagrams, which is the rank.
General and special linear groups.
The general linear group GL"n"("R") is the group of all "R"-linear automorphisms of "R""n". There is a subgroup: the special linear group SL"n"("R"), and their quotients: the projective general linear group PGL"n"("R") = GL"n"("R")/Z(GL"n"("R")) and the projective special linear group PSL"n"("R") = SL"n"("R")/Z(SL"n"("R")). The projective special linear group PSL"n"("F") over a field "F" is simple for "n" ≥ 2, except for the two cases when "n" = 2 and the field has order 2 or 3.
Unitary groups.
The unitary group U"n"("R") is a group preserving a sesquilinear form on a module. There is a subgroup, the special unitary group SU"n"("R") and their quotients the projective unitary group PU"n"("R") = U"n"("R")/Z(U"n"("R")) and the projective special unitary group PSU"n"("R") = SU"n"("R")/Z(SU"n"("R"))
Symplectic groups.
The symplectic group Sp2"n"("R") preserves a skew symmetric form on a module. It has a quotient, the projective symplectic group PSp2"n"("R"). The general symplectic group GSp2"n"("R") consists of the automorphisms of a module multiplying a skew symmetric form by some invertible scalar. The projective symplectic group PSp2"n"(F"q") over a finite field is simple for "n" ≥ 1, except for the cases of PSp2 over the fields of two and three elements.
Orthogonal groups.
The orthogonal group O"n"("R") preserves a non-degenerate quadratic form on a module. There is a subgroup, the special orthogonal group SO"n"("R") and quotients, the projective orthogonal group PO"n"("R"), and the projective special orthogonal group PSO"n"("R"). In characteristic 2 the determinant is always 1, so the special orthogonal group is often defined as the subgroup of elements of Dickson invariant 1.
There is a nameless group often denoted by Ω"n"("R") consisting of the elements of the orthogonal group of elements of spinor norm 1, with corresponding subgroup and quotient groups SΩ"n"("R"), PΩ"n"("R"), PSΩ"n"("R"). (For positive definite quadratic forms over the reals, the group Ω happens to be the same as the orthogonal group, but in general it is smaller.) There is also a double cover of Ω"n"("R"), called the pin group Pin"n"("R"), and it has a subgroup called the spin group Spin"n"("R"). The general orthogonal group GO"n"("R") consists of the automorphisms of a module multiplying a quadratic form by some invertible scalar.
Contrast with exceptional Lie groups.
Contrasting with the classical Lie groups are the exceptional Lie groups, G2, F4, E6, E7, E8, which share their abstract properties, but not their familiarity. These were only discovered around 1890 in the classification of the simple Lie algebras over the complex numbers by Wilhelm Killing and Élie Cartan.


Grassmann product


Kanta Gupta

Chander Kanta Gupta (8 October 1938 – 27 March 2016) was a Canadian distinguished professor of mathematics at the University of Manitoba, known for her research in abstract algebra and group theory. Much of her research concerns the automorphisms in different varieties of groups.
Education.
Gupta earned a bachelor's degree from the University of Jammu and Kashmir, a master's degree from the Aligarh Muslim University, another master's degree from the Australian National University, and a Ph.D. in 1967 from ANU under the supervision of Michael Frederick Newman.
Recognition.
She was elected to the Royal Society of Canada in 1991, and awarded the Krieger–Nelson Prize of the Canadian Mathematical Society in 2000.
Personal.
Her husband, Narain Gupta (1936–2008) was also an ANU alumnus, and a distinguished professor of mathematics at the University of Manitoba.


Butson-type Hadamard matrix

In mathematics, a complex Hadamard matrix "H" of size "N" with all its columns (rows) mutually orthogonal, belongs to the Butson-type "H"("q", "N") if all its elements are powers of "q"-th root of unity,
Existence.
If "p" is prime and formula_2, then formula_3 can exist
only for formula_4 with integer "m" and
it is conjectured they exist for all such cases
with formula_5. For formula_6, the corresponding conjecture is existence for all multiples of 4.
In general, the problem of finding all sets
formula_7 such that the Butson - type matrices
formula_8 exist, remains open.
Examples.
belong to the Butson-type,