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The Hopf modules category and the Hopf equation

We study the Hopf equation which is equivalent to the pentagonal equation, from operator algebras. A FRT type theorem is given and new types of quantum groups are constructed. The key role is played now by the classical Hopf modules category. As an application, a five dimensional noncommutative noncocommutative bialgebra is given.


Introduction
Let H be a bialgebra over a field k. There are two fundamental categories in the theory of Hopf algebras and quantum groups: H M H , the category of H-Hopf modules and H YD H , the category of quantum Yetter-Drinfel'd modules. The objects in these categories are k-vector spaces M which are left H-modules (M, ·), right H-comodules (M, ρ), such that the following quite distinct compatibility relations hold: in the case H M H , and respectively for the Yetter-Drinfel'd categories.
However, there are two connections between these categories. The first one was given by P. Schauenburg in [22]: it was proven that the category H YD H is equivalent to the category H H M H H of two-sided, two-cosided Hopf modules. The second was given recentely in [4]. For A an H-comodule algebra and C an H-module coalgebra, Doi (see [7]) and independently Koppinen (see [12]) defined A M(H) C , the category of Doi-Koppinen Hopf modules, whose objects are left A-modules and right C-comodules and satisfy a compatibility relation which generalises (1). In [4] it was proven that H YD H is isomorphic to H M(H op ⊗ H) H , where H can be viewed as an H op ⊗ H-module (comodule) coalgebra (algebra). The isomorhism is the identity functor M → M. We hereby obtain a strong link between the categories H M H and H YD H : both are particular cases of the same general category A M(H) C . This led us in [5], [6] to study the implications of the category H YD H in the classic, non-quantic part of Hopf algebra theory. In [5] we start with the following classic theorem (see [19]): any finite dimensional Hopf algebra is Frobenius. In the language of categories, this result is interpreted as follows: the forgetful functor H M H → H M is Frobenius (i.e., cf. [5], by definition has the same left and right adjoint) if and only if H is finite dimensional. The next step is easy to take: we must generalize this result for the forgetful functor A M(H) C → A M and then apply it in the case of Yetter-Drinfel'd modules for the forgetful functor H YD H → H M. We thus obtain the fact that the forgetful functor H YD H → H M is Frobenius if and only if H is finite dimensional and unimodular (see theorem 4.2 of [5]). The same treatment was applied in [6] for the classic Maschke theorem. One of the major obstacles was to correctly define the notion of integral for the Doi-Hopf datum (H, A, C), such as to be connected to the classic integral on a Hopf algebra (corresponding to the case C=A=H), as well as to the notion of total integral (corresponding to the case C=A) defined by Doi in [8]. This technique can be looked upon as a "quantisation" of the theorems from the classic theory of Hopf algebras. There are two steps to it: first, we seek to generalize a result for the category A M(H) C , then to apply it to the particular H YD H case. There is also another approach to this "quantisation" technique, recently evidenced in [9] for the same Frobenius type theorem. It was first proven, by generalizing the classic result, that any finite dimensional Hopf algebras extensions is a β-Frobenius extension (or a Frobenius extension of second kind). Then, this theorem was "quantised" to the case of Hopf algebras extensions in H YD H . The result includes the case of enveloping algebras of Lie coloralgebras.
Beginning with this paper, we shall tackle the reverse problem: we shall try to involve the category H M H in fields dominated until now by H YD H , i.e. try a "dequantisation". For the beginning, it is enough to remind that the category H YD H is deeply involved in the quantum Yang-Baxter equation: where R ∈ End k (M ⊗ M), M being a k-vector space. The starting point of this paper is the following question: An interesting connection between the pentagonal and the quantum Yang-Baxter equations is given in [30]. The pentagonal equation plays a fundamental role in the duality theory for operator algebras (see [2] and the references indicated here). If H is a Hopf algebra, then is a bijective solution of the Hopf equation. Moreover, the comultiplication ∆ can be rebuilt from R by means of where z ∈ H ⊗ H such that R(z) = 1 ⊗ 1. This operator was defined first by Takesaki in [28] for a Hopf-von Neumann algebra. The operator W := τ Rτ is called in [15] the evolution operator for a Hopf algebra and plays an important role in the description of the Markov transition operator for the quantum random walks (see [15] or [16] Obviously, substituting the key map R (M,·,ρ) with τ R (M,·,ρ) τ , all results of this paper remain valid if we replace the Hopf equation with the pentagonal equation. We have preferred however to work with the Hopf equation, for historical reasons: this is how the issue has been raised for the first time in lemma 4.9 of [28].
This study was continued in [18], where new classes of bialgebras arising from the Hopf equation are introduced and analyzed.

Preliminaries
Throughout this paper, k will be a field. All vector spaces, algebras, coalgebras and bialgebras that we consider are over k. ⊗ and Hom will mean ⊗ k and Hom k . For a coalgebra C, we will use Sweedler's Σ-notation, that is, etc. We will also use Sweedler's notation for right C-comodules: is a right C-comodule. M C will be the category of right Ccomodules and C-colinear maps and A M will be the category of left A-modules and A-linear maps, if A is a k-algebra.
Recall the following well known lemmas: Lemma 1.1 Let M be a finite dimensional vector space with {m 1 , · · · , m n } a basis for M and let C be a coalgebra. We define the k-linear map ρ : M → M ⊗C, ρ(m l ) = n v=1 m v ⊗c vl , for all l = 1, · · · , n, where (c vl ) v,l is a family of elements of C. The following statements are equivalent: for all j, k = 1, · · · , n If we denote B = (c vl ) v,l , then, as usual, the relations (5)  Let H be a bialgebra. Recall that an (left-right) H-Hopf module is a left H-module (M, ·) which is also a right H-comodule (M, ρ) such that will play an important role in the present paper. It is useful to point out the following lemma. The proof is left to the reader. For a vector space V , τ : V ⊗V → V ⊗V will denote the switch map, that is, Using the notation R(u ⊗ v) = u 1 ⊗ v 1 then where the subscript (0) means that w is not affected by the application of R 12 .
Let H be a bialgebra and (M, ·) a left H-module which is also a right H-comodule (M, ρ).
Recall that (M, ·, ρ) is a Yetter-Drinfel'd module if the following compatibility relation holds: for all h ∈ H, m ∈ M. H YD H will be the category of Yetter-Drinfel'd modules and H-linear H-colinear homomorphism. If (M, ·, ρ) is a Yetter-Drinfel'd module then the special map R = R (M,·,ρ) given by the equation (7) is a solution of the quantum Yang-Baxter equation If M is a finite dimensional vector space and R is a solution of the quantum Yang-Baxter equation, then there exists a bialgebra A(R) such that (M, ·, ρ) ∈ A(R) M A(R) and R = R (M,·,ρ) (see [20]). For a further study of the Yetter-Drinfel'd category we refer to [14], [20], [21], [31], or to the more recent [4], [5], [6], [9].

The Hopf equation
We will start with the following Definition 2.1 Let V be a vector space and R ∈ End k (V ⊗ V ).

We shall say that R is a solution for the pentagonal equation if
Remarks 2.2 1. The Hopf equation is obtained from the quantum Yang-Baxter equation by deleting the midle term from the right hand side.
2. Let {m i } i∈I be a basis of the vector space V . Then an endomorphism R of V ⊗ V is given by a family of scalars (x kl ij ) i,j,k,l∈I of k such that for all v, u ∈ I. A direct computation shows us that R is a solution of the Hopf equation if and only if (x kl ij ) i,j,k,l∈I is a solution of the nonlinear equation for all j, k, l, u, v, w ∈ I. It follows that solving the system (11) is really a non-trivial problem.
3. Using the notation R(x ⊗ y) = x 1 ⊗ y 1 , for x, y ∈ V , then R is a solution of the Hopf equation if and only if

Let
A be an algebra and R ∈ A ⊗ A be an invertible element such that the Hopf equation for all a ∈ A is coassociative and an algebra map. Indeed, for all a ∈ A. But, as R satisfies the Hopf equation, we have that W = R 12 , i.e. equation (12) holds.
In the next proposition we shall evidence a few equations which are equivalent to the Hopf equation. Proof 1 ⇔ 2 The proof will follows from the formulas: and from the fact that τ 13 is an automorphism of and 1 ⇔ 3 follows from the formulas: 1 ⇔ 4 follows from the formulas:

⊓ ⊔
From now on we shall study only the Hopf equation. In [2] numerous examples of operators W which are solutions for the pentagonal equation are given. All these operators come from the theory of operator algebras. We shall present only one of them, which plays a key role in classifying the multiplicative and commutative operators (see theorem 2.2 from the above cited paper). Let G be a locally compact group and dg a right Haar measure on G. Then, hence u R is also a solution of the Hopf equation.
3. Let f , g ∈ End k (V ) such that f 2 = f , g 2 = g and f g = gf . Then, R := f ⊗ g is a solution of the Hopf equation.
A direct computation shows that so the above conclusion follows. With this example in mind we can view the Hopf equation as a natural generalization of the idempotent endomorphism. That because R = f ⊗ I (or such that with respect to the given basis is where q is a scalar of k. Then f 2 q = f q . Let g q ∈ End k (V ), g q = Id V − f q . Then g q is also an idempotent endomorphism of V and g q f q = f q g q . Thus, we obtain that R q = f q ⊗ g q with respect to the basis and R q is a solution for the Hopf equation.
Then with respect to the same ordonate basis of V ⊗ V , R ′ q is given by and R ′ q is also a solution of the Hopf equation. 4. The above R q and R ′ q are also solutions of the quantum Yang-Baxter equation, because each of them has the form f ⊗ g with f g = gf . In this example we will construct a solution for the Hopf equation which is not a solution of the quantum Yang-Baxter equation.
Let G be a group and V be a G-graded representation on G, that is V is a left k[G]-module and there exists {V σ | σ ∈ G} a family of subspaces of V such that is a solution of the Hopf equation and is not a solution of the quantum Yang-Baxter equation.
Indeed, it is enought to prove that (8) holds only for homogenous elements. Let u σ ∈ V σ , u τ ∈ V τ and u θ ∈ V θ . Then, Hence R is a solution of the Hopf equation. On the other hand, by a direct computation we get is not a solution of the quantum Yang-Baxter equation.
5. Let G be a group and V be a G-crossed module, that is V is a left k[G]-module and there exists {V σ | σ ∈ G} a family of subspaces of V such that for all g, σ ∈ G. Then R given by (14) is a solution of the quantum Yang-Baxter equation and is not a solution of the Hopf equation.
6. Let q be a scalar of k, q = 0, q = 1. Then the classical two dimensional Yang-Baxter operator is a solution of the Yang-Baxter equation and is not a solution for the Hopf equation.
Indeed, the element in the (1, 1)-position of R 23 R 13 R 12 is q 3 , while the element in the (1, 1)position of R 12 R 23 is q 2 , i.e. R is not a solution of the Hopf equation.

Let H be a bialgebra. Then
for all g, h ∈ H, is a solution of the Hopf equation. This operator was defined by Takesaki in [28] for a Hopf-von Neumann algebra (A, ∆).
8. Let H be a Hopf algebra with an antipode S. Then H/k is a Hopf-Galois extension (see [19]), i.e. the canonical map is bijective. Then β is a solution of the Hopf equation. Furthermore, is also a solution of the Hopf equation.
In [2], the concept of multiplicative and commutative (respectively cocommutative) operator is introduced: that is, a unitary operator W ∈ L(K ⊗K), where K is a Hilbert space, W satisfies the pentagonal equation and We shall now introduce the corresponding concept for the Hopf equation.  we get that the equation (15) is equivalent to τ 12 τ 23 W 13 W 23 τ 12 τ 13 = τ 13 τ 12 W 23 W 13 τ 23 τ 12 .

Suppose that
i.e. R is a solution of the Hopf equation.

We have
As H is commutative, we obtain that R 12 R 13 = R 13 R 12 .

A FRT type construction for Hopf modules
In this section we shall prove the main result of the paper, which shows us that in the finite dimensional case any solution of the Hopf equation has the form R (M,·,ρ) . Proof 1. The proof will be given is several steps. Let {m 1 , · · · , m n } be a basis for M and (x ji uv ) i,j,u,v a family of scalars of k such that for all u, v = 1, · · · , n.
Let (C, ∆, ε) = M n (k), be the comatrix coalgebra of order n, i.e. C is the coalgebra with the basis {c ij | i, j = 1, · · · , n} such that for all j, k = 1, · · · , n. Let ρ : M → M ⊗ C given by for all l = 1, · · · , n. Then, by lemma 1.1, M is a right C-comodule. Let T (C) be the bialgebra structure on the tensor algebra T (C) which extends ∆ and ε (from lemma 1.
for all j, u, v = 1, · · · , n. For m v , m u the elements of the given basis, we have: Hence, (M, ·, ρ) has a structure of left T (C)-module and right T (C)-comodule such that R = R (M,·,ρ) . Now, we define the obstructions χ(i, j, k, l) which measure how far away M is from a T (C)-Hopf module. Keeping in mind that T (C) is generated as an algebra by (c ij ) and using lemma 1.3 we compute only for h = c jk , and m = m l , for j, k, l = 1, · · · , n. We have: for all i, j, k, l = 1, · · · , n. Then Let I be the two-sided ideal of T (C) generated by all χ(i, j, k, l), i, j, k, l = 1, · · · , n. The key point of the proof follows: I is a bi-ideal of T (C) and I · M = 0.
We first prove that I is also a coideal and this will result from the following formula: Indeed, we have: where in the last equality we use the fact that a,b,γ x jγ ab c iγ ⊗ c ak c bl = p,r,s x jp rs c ip ⊗ c rk c sl Hence, the formula (21) holds. On the other hand ε χ(i, j, k, l) = x ji kl − x ji kl = 0 so we proved that I is a coideal of T (C). Now, in order to show that I · M = 0, we shall use the fact that R is a solution of the Hopf equation. For z ∈ M, j, k = 1, · · · , n, we have the following formula: Let us compute On the other hand It follows that From the universal property of the tensor algebra T (C), there exists a unique algebra map f 1 : T (C) → H such that f 1 (c ij ) = c ′ ij , for all i, j = 1, · · · , n. As (M, · ′ , ρ ′ ) ∈ H M H we get that χ ′ (i, j, k, l) = 0, and hence f 1 (χ(i, j, k, l)) = 0, for all i, j, k, l = 1, · · · , n. So the map f 1 factorizes to the map f : Of course, for m l an arbitrary element of the given basis of M, we have Conversely, the relation (I ⊗ f )ρ = ρ ′ necessarily implies f (c ij ) = c ′ ij , which proves the uniqueness of f . This completes the proof of the theorem.
3. For z ∈ M and j, k = 1, · · · , n we have the formula: Let I be the two-sided ideal of T (C) generated by I and all [c rk , c sj ]. It follows from the formula that I is also a coideal of T (C) and from equation (23)  Our obstruction elements χ(i, j, k, l) play the same role as the homogenous elements d(i, j, k, l) defined in [20] which correspond to the quantum Yang-Baxter equation: the two-sided ideal generated by them is also a coideal which anihilates M. This was the key point of the proof.
2. The last point of our theorem can by viewed as an algebraic version of theorem 2.2. from [2]. All commutative bialgebras B(R) are quotients for various bialgebra structures which can be given on k[Y 1 , · · · , Y n ].

Applications
In this section we shall construct new examples of noncommutative noncocommutative bialgebras arising from our FRT type theorem. As the relations through which we factor are not all homogenous, all our examples are different from the ones which appear in quantum group theory. A completely different method for constructing such objects uses Ore extensions and was recently evidenced in [3].
In the next propositions of this section, the relations χ(i, j, k, l) = 0 will be written in the lexicografical order according to (i, j, k, l) starting with (1, 1, 1, 1).

Back to euclidian geometry
There exists an intimate link between the quantum Yang-Baxter equation and the quantum plane k q < x, y | xy = qyx > (see [13]).
We shall now show that the term "dequantisation" used in the introduction is not an abuse. More specifically, our FRT type construction for the Hopf equation supplies us with a way to construct noncommutative and noncocommutative bialgebras starting from the elementary maps of plane euclidian geometry: projections of k 2 on the Ox and Oy axis. The map f 0 , corresponding to q = 0 in the equation (13), is in fact the projection of the plane k 2 on the Ox axis. We can now associate three bialgebras to this projection: the first one corresponds to the solution of the Hopf equation f 0 ⊗ g 0 , where g 0 = Id k 2 − f 0 , i.e. is the projection of the plane k 2 on the Oy axis, the second corresponds to f 0 ⊗ Id k 2 , and the third correspunds to the f 0 ⊗ f 0 . In this way, we obtain the bialgebras denoted below by B 2 0 (k), D 2 0 (k) and E 2 0 (k). If q = 0 and k = R, the map f q given in (13) sends all the points of the R 2 plane on the Ox axis under an angle arctg(q) with respect to the Oy axis. Correspondingly, the bialgebras B 2 q (k), D 2 q (k) and E 2 q (k) are constructed.

Proposition 4.1 Let q be a scalar of the field k and R q be the solution of the Hopf equation given by
. Then: 1. If q = 0, the bialgebra B 2 0 (k) is the free algebra generated by x, y, z with the relations: The comultiplication ∆ and the counity ε are given by: 2. If q = 0, the bialgebra B 2 q (k) is the free algebra generated by A, B with the relation: The comultiplication ∆ and the counity ε are given by: Proof Let M be a two dimensional vector space with {m 1 , m 2 } a basis. Put R = R q . With respect to the ordonate basis {m 1 ⊗ m 1 , m 1 ⊗ m 2 , m 2 ⊗ m 1 , m 2 ⊗ m 2 }, R is given by: x ji uv m i ⊗ m j we get that among the elements (x ji uv ), the only nonzero elements are: Hence, c 21 = 0. Now, if we denote c 11 = x, c 22 = y, c 12 = z, there are only two linear independent relations: yx = x, yz + qy 2 = qx.
If q = 0, then follows 1. If q = 0, then x is an element in the free algebra generated by y and z. Let A = y and B = z + qy = z + qA. Then and by substituting in the first relation we get A 2 B = AB. The formulas for ∆ and ε follow as the original (c ij ) was a comultiplicative matrix. ⊓ ⊔

Remark 4.2
The bialgebra B 2 0 (k) is not a Hopf algebra. We can localize it to obtain a Hopf algebra. As ∆(x) = x ⊗ x, ∆(y) = y ⊗ y and ε(x) = ε(y) = 1 we should add new generators which make x and y invertible. But then y = 1 and z = 0. It follows that if we localize the bialgebra B 2 0 (k), we get the usual Hopf algebra k[X, X −1 ], with ∆(X) = X ⊗ X, ε(X) = X ⊗ X, and with the antipode S(X) = X −1 . We denote by D 2 q (k) the bialgebra B(R ′ q ). Then: 1. If q = 0, the bialgebra D 2 0 (k) is the free algebra generated by x, y, z with the relations: The comultiplication ∆ and the counity ε are given by: 2. If q = 0, the bialgebra D 2 q (k) is the free algebra generated by A, B with the relations: The comultiplication ∆ and the counity ε are given by: Proof We start exactly as in the above proposition. We get that among the scalars (x ji uv ), which define R, the only nonzero elements are: Hence c 21 = 0. If we denote c 11 = x, c 22 = y, c 12 = z then we get the following six relations: xz + qxy = yz + qy 2 = qx.
Using the fact that y(y − 1) = (y − 1)y, the fifth relation follows from the second and the fourth. On the other hand, as y(y 2 − 1) = (y + 1)(y − 1)y, the first relation follows from the second and the third. We have thus reduced the above five relations to only three: (y − 1)y(z + qy) = 0, zy(z + qy) = 0, z(z + qy) = 0.
Further, let B = z and A = q −1 (z + qy) = q −1 (B + qy). It follows that y = A − q −1 B. The third relation takes the form BA = 0 and this implies the second one. Using BA = 0, the first relation becomes A 3 = A 2 . The formulas for ∆ and ε follow from equation (24). We note that the element A − q −1 B is a groupal element of D 2 q (k). ⊓ ⊔ Remark 4.4 R ′ q is also a solution of the quantum Yang-Baxter equation. The bialgebra A(R ′ q ) which we obtain applying the usual FRT construction is the free algebra generated by x, y, z, t with the relations: zx = xz = zy = z 2 = zt = 0, xy − yx = qyz, xt − tx = qtz, xy + qxt = qx 2 , y 2 + qyt = qxy, ty + qt 2 = qxt.
The comultiplication ∆ and the counity ε are given in such way that the matrix x y z t is comultiplicative.
In the next proposition we shall prove that the bialgebra E 2 q (k), with q = 0, which can be associated to the solution R ′′ q = f q ⊗ f q is not dependent of q, i.e. E 2 q (k) ∼ = E 2 q ′ (k), for all q, q ′ ∈ k\{0}. Proposition 4.5 Let q be a scalar of the field k and R ′′ q be the solution of the Hopf equation given by . Then: 1. If q = 0, the bialgebra E 2 0 (k) is the free algebra generated by x, y, z with the relations: The comultiplication ∆ and the counity ε are given by: 2. If q = 0, the bialgebra E 2 q (k) is the free algebra generated by A, B with the relations: The comultiplication ∆ and the counity ε are given by: Proof We start exactly as in the above propositions. We get that among the scalars (x ji uv ), which define R, the only nonzero elements are: x 2 = x, xz + qxy = qx, zx + qyx = qx, So, if q = 0, we obtain the relations of E 2 0 (k). If q = 0, then x is in the free algebra generated by y and z and x = y 2 + q −1 zy + q −1 yz + q −2 z 2 = (y + q −1 z) 2 .
If we substitute x in the other three relations, only remains, the other two being linear dependent from this one. Now, if we denote A = y and B = y + q −1 z, we obtain the description of E 2 q (k). ⊓ ⊔ Remarks 4.6 1. If q = 0, it is interesting to denote that B 2 is a groupal element of E 2 q (k). Indeed, we have 2. The bialgebra D 2 0 (k) is the quotient of B 2 0 (k) by the two-sided ideal (which is also a coideal) generated by x 2 − x, zx, xz, z 2 .
is also a quotient of E 2 0 (k) by the two-sided ideal generated by yx − x, yz.
• The comultiplcation ∆ and the counity ε are given in such a way that the matrix (c ij ) i,j is comultiplicative.
The proof is similar to the one of proposition 4.1. Among the elements (x ji uv ), which define π 1 ⊗ π 1 , the only nonzero elements are If i = 1, all the relations χ(i, j, k, l) = 0 are 0 = 0, with the exception of the relations χ(i, j, j, 1) = 0 for all j ≥ 2, which give us 0 = c i1 for all i ≥ 2. If i = 1 the relations χ(1, j, k, l) = 0 give us c jk c 1l = δ kj δ l1 c 11 for all j ≥ 2 and k, l ≥ 1.
New types of bialgebras can be constructed starting from projections of k n on different intersections of hyperplanes.

A five dimensional noncommutative noncocommutative bialgebra
Let p be a prime number. The classification of p dimensional Hopf algebras over a field of positive characteristic is still an open problem (we remind that in [32] Zhu proved that, over an algebraically closed field of characteristic zero, any p dimensional Hopf algebra is isomorphic to the groupal algebra k[Z p ]). Classifying the p dimensional bialgebras seems to be a much more complicated problem.

Remark 4.7
We notice that y does not appear in the relations of E 2 0 (k). As ∆(y − 1) = (y − 1) ⊗ y + 1 ⊗ (y − 1) and ε(y − 1) = 1, we get that the two-sided ideal generated by y − 1 is also a coideal. We can add the new relation y = 1 in the definition of E 2 0 (k) and we obtain a three dimensional noncocommutative bialgebra. We denote this bialgebra with T (k). Then: • As a vector space, T (k) is three dimensional with {1, x, z} a k-basis.
• The multiplication rule is given by: • The comultiplication ∆ and the counity ε are given by In [11], over a field of characteristic two, two examples of three dimensional bialgebras are given. Both of them are commutative and cocommutative. Our T (k) differs from one of them only by the relation ∆(z) = x ⊗ z + z ⊗ 1 (respectively ∆(z) = 1 ⊗ z + z ⊗ 1 in [11]). This minor change of ∆ (in our case k being a field of arbitrary characteristic) makes T (k) noncocommutative.
T (k) * is a three dimensional noncommutative bialgebra. It follows that T (k) ⊗ T (k) * is a nine dimensional noncommutative and noncocommutative bialgebra.
If in the preceding remark we have constructed a noncocommutative but commutative three dimensional bialgebra, now we shall construct a special R such that B(R) is a five dimensional noncommutative noncocommutative bialgebra. 2. If char(k) = 2, then the bialgebra B(R) is the free algebra generated by x, y, z with the relations: The comultiplication ∆ and the counity ε are given by: Furthermore, dim k (B(R)) = 5.
Proof 1. By a direct computation we obtain: It follows that: where α = 1 + 1. Hence, R is a solution of the Hopf equation if and only if char(k) = 2. In this case we also have that i.e. R is commutative.
2. Suppose that char(k) = 2. Starting as in the proof of the above propositions, with a basis in a two dimensional vector space, we obtain that among (x kl ij ) the only nonzero elements are Now, if we denote c 11 = x, c 12 = y, c 21 = z, c 22 = t and using that char(k) = 2 we get the following relations: x 2 = x, y 2 = z 2 = yx = yz = 0, xy = y, xz = zx = z, zy = x + t, t 2 = t, xt = t, tx = x, yt = 0, ty = y, zt = tz = z.
So, t is in the free algebra generated by x, y, z and t = zy − x.
If we substitute t in all the relations in which t is involved, then these become identities. The relations given in the statement of the proposition remain. The formula for ∆ follows, as the matrix x y z t was comultiplicative.
All these relations give us that {1, x, y, z, zy} is a sistem of generators of B(R) as a vector space over k. It remain to check that they are liniar independent over k. Let a, b, c, d, e ∈ k such that a + bx + cy + dz + e(zy) = 0.
First, we multiply to the left with y and we get a = 0. Then we multiply to the right with z and we obtain that b = 0. Now we multiply to the right with x and we get d = 0. If we multiply now to the left with z we get c = 0, and e = 0 follows. Hence {1, x, y, z, zy} is a k-basis for B(R). ⊓ ⊔ Remarks 4.9 1. The bialgebra B(R) constructed in the above proposition is not a Hopf algebra. We can localize B(R) and we get a Hopf algebra which is isomorphic to the grupal Hopf algebra k[G], where G is a group with two elements. Indeed, let S be a potential antipode. Then: S(x)x + S(y)z = 1, S(x)y + S(y)t = 0.
2. Directly from the proof, we obtain an elementary definition for the bialgebra B(R): • As a vector space B(R) is five dimensional with {1, x, y, z, t} a k-basis.
• The comultiplcation ∆ and the counity ε are given in such way that the matrix x y z t is comultiplicative.
4. Recently, in [10], considering quotients of the bialgebras B(R) for various solutions R of the Hopf-equation, numerous examples of finite dimensional noncommutative and noncocommutative bialgebras are constructed. We shall present one of them, which is a quotient of E 2 0 (k). Let n ≥ 2 be a natural number. Then, the two-sided ideal I of E 2 0 (k) generated by y n − y, zy, xy − x and yx − x is a biideal and B 2n+1 (k) := E 2 0 (k)/I is a 2n + 1-dimensional noncommutative noncocommutative bialgebra (see [10] for the proof). The bialgebra B 2n+1 (k) can be described as follows: • B 2n+1 (k) is the free algebra generated by x, y, z with the relations: x 2 = x, xz = zx = z 2 = zy = 0, y n = y, xy = yx = x.