On the category of modules over some semisimple bialgebras

We study the tensor category of modules over a semisimple bialgebra H under the assumption that irreducible H-modules of the same dimension > 1 are isomorphic. We consider properties of Clebsch–Gordan coefficients showing multiplicities of occurrences of each irreducible H-module in a tensor product of irreducible ones. It is shown that, in general, these coefficients cannot have small values.

A semisimple algebra H is a direct sum of full matrix algebras over k. One-dimensional summands are in one-to-one correspondence with algebra homomorphisms H → k. Hence, under our assumptions, H as a k-algebra has a semisimple direct decomposition Mat(d j , k) ⎞ ⎠ , (1.1) where n, d j are natural numbers and {e g , g ∈ G} is a system of central orthogonal idempotents in H corresponding to the one-dimensional direct summands. For h ∈ H and g ∈ G we have he g = e g h = g, h e g . As in [1], we here deal with the case when which just means that irreducible H -modules of the same dimension > 1 are isomorphic.
The main result of the paper [1] is the following: In the present paper, for a bialgebra H, we consider properties of the Clebsch-Gordan coefficients, that is, the multiplicities of occurrences of irreducible H -modules in semisimple decompositions of tensor products of irreducible ones. These play a substantial role in representation theory of groups and their applications to physics.
More general than in [1], we consider the case of a bialgebra H not assuming that it is a Hopf algebra. In Theorem 4.5, under some restrictions on the Clebsch-Gordan coefficients, it is shown that n 2 in (1.1). In Theorem 4.6, for the case n = 2, we compare the number of one-dimensional summands in (1.1) and the sizes of matrix components. Further properties of Clebsch-Gordan coefficients are found in Theorem 4.7. In the last section we consider the comodule structure of H.

Bialgebra structure of H and H *
We consider comultiplication and counit in the bialgebra H having as algebra a decomposition (1.1). The counit ε : H → k has the form For each one-dimensional H -module E g = ke g related to g ∈ G, For further information on the bialgebra structure of H some additional properties of the dual bialgebra H * are needed. The semisimple bialgebra H over an algebraically closed field k has the decomposition (1.1). If char k = 0 and H is a Hopf algebra, then, by the Larson-Radford theorem [4,Theorem 7.4.6], the dual Hopf algebra H * is also semisimple. Recall that some additional information on semisimple Hopf algebras in positive characteristic can be found in [6].
Consider one of the main samples of bialgebras, namely a monoid algebra F = kG of a finite monoid G. In this case (g) = g ⊗ g for any g ∈ G. It means that G is the monoid of group-like elements of F.
It is well-known that the dual bialgebra F * is a direct sum of one-dimensional ideals ⊕ g∈G ke g . Here {e g | g ∈ G} is the dual base for the base {g | g ∈ G} of F. In particular, F * is semisimple.
However, its dual bialgebra F * * = F is not necessarily semisimple. For example, take the three-element commutative monoid G = {1, a, b} with the identity element 1 such that ab = b 2 = a 2 = b. Then the one-dimensional space k(a − b) in the monoid algebra F = kG is annihilated by a, b. Hence it is a nilpotent ideal and the monoid algebra kG is not semisimple.
We shall now expand these structural observations to the case of the bialgebra H from (1.1).
Consider in each matrix component Mat(d i , k), the non-degenerated symmetric bilinear form In the case of a Hopf algebra we consider the form x, y = tr(x · S(y)) where S is the antipode [3]. We shall prove results from [3, Section 3] on Hopf algebras for the bialgebra case. Using the form (2.3), we can identify the space Thus, as a vector space, H * has a direct decomposition

Proposition 2.1
The following conditions are satisfied: and this means * (E Proof Suppose that a is from (2.4). Then by (2.6) and therefore p * q = pq.
In the case of Hopf algebras we can prove the last formulas as in [3].
Now we shall consider some new properties of the bialgebra H from (1.1). The bialgebra H is a left and right H * -module algebra with respect to actions f x, For f ∈ G, the maps f , f are algebra endomorphisms of H preserving the identity element 1 of H, As shown in [ With respect to the natural pairing −, − , the elements g ∈ G ⊂ H * are dual to the elements e g , g ∈ G, and each matrix component is annihilated by elements of G. (3) By (2), the element g e 1 = 0 if and only if there exists an element p ∈ G such that pg = 1. It means that p = g −1 .
(4) Let g ∈ G. The map h → (g h) is an algebra endomorphism of H preserving the unit element k). Each full matrix algebra Mat(d i , k) is simple and therefore it is mapping either to zero or injectively into H. Hence we obtain the required equality by (2).

Theorem 2.4 Let α be a unit preserving endomorphism of the semisimple algebra R
Suppose that each integer d j is not a linear combination of d 1 , . . . , d j−1 with non-negative integer coefficients. Then α is an automorphism of R preserving each matrix component. Proof We shall proceed by induction on n. If n = 1, then α is an endomorphism of the full matrix algebra preserving the unit element. Hence α is injective and therefore it is surjective.
Suppose that the theorem is proved for n − 1. Since d n > d j for any j < n we can conclude that Mat(d n , k) is stable under α. By induction, α induces an automorphism on R/ Mat(d n , k). So without loss of generality we can assume that α is identical modulo Mat(d n , k). It means that if x ∈ Mat(d j , k), j < n, k) is an algebra homomorphism, not necessarily preserving the unit element.
Suppose first that α(E (n) ) = 0. Then α induces an automorphism of Mat(d n , k) and therefore α(E (n) ) = E (n) . If x ∈ Mat(d j , k), j < n then x E (n) = 0 in R and therefore Hence, in this case, α is an automorphism and the proof is complete.
By the Noether-Skolem and centralizer theorems, we can conclude that Mat(t j , k) β j (Mat(d j , k)) ⊗ Mat(s j , k) for some non-negative integer s j . Hence t j = d j s j and therefore d n = t 1 + · · · + t n−1 = d 1 s 1 + · · · + d n−1 s n−1 , a contradiction.
Note that the restriction on the numbers in Theorem 2.4 is satisfied if, for each j, the greatest common divisor of d 1 , . . . d j is smaller than the greatest common divisor of d 1 , . . . , d j−1 .

The category of modules
Let H be, as above, a semisimple bialgebra with direct sum decomposition (1.1) such that (1.2) is satisfied. In what follows we shall in addition assume that either G is a group or d 1 , . . . , d n are as in Theorem 2.4. In both cases, for each g ∈ G, the map g induces an algebra automorphism of every matrix component in (1.1). The tensor product M ⊗ N of two left H -modules M, N is again a left H -module by putting, for h ∈ H and (h) = h h (1) ⊗ h (2) , (1) x ⊗ h (2) y, x ∈ M, y ∈ N . (3.1) Let M i be the irreducible H -module associated with matrix component Mat(d i , k). The module M i is annihilated by each element e g , g ∈ G, and by any Mat(d j , k), j = i.
Note that if h ∈ Mat(d i , k) and x ∈ M p , y ∈ M q , then by (3.1) we have where i pq (h) · (x ⊗ y) is the componentwise action on the tensor product. As in [1, Formula (9), Lemma 3.1] we can prove: Proposition 3.1 Let h ∈ H, g ∈ G and D g,i from (2.6). If x, y ∈ M i then h(D g,i ·(x⊗y)) = g, h D g,i ·(x⊗y) and D 2 g,i = D g,i .
Proof We have The last statement holds because e g is an idempotent.
The next fact is well known for Hopf algebras [1]. In virtue of Theorem 2.3 it holds for bialgebras H satisfying the above restrictions. For any square matrix X denote its transpose by t X. Let M i be as above the irreducible H -module of dimension d i Then the dual space

Proposition 3.2 Let H be a bialgebra with a direct decomposition
The next affirmation follows from associativity of tensor products of H -modules. Denote by R t , 1 t n, the square matrix of size n whose (i, j)th entry is equal to m t i j . Then R r is a non-negative integer matrix. By Theorem 3.6, each matrix R t is symmetric. Now the equality (3.4) and the statement of Theorem 3.6 can be rewritten as where E n and E l j are the identity matrix and the matrix units of size n. If H is a Hopf algebra, then each matrix R i is symmetric. which all are non-negative integers. Then Now the first equation in (3.5) can be rewritten as b 2 + c 2 − ac − bd = |G|, (3.8) and the second equation in (3.5) as (3.9)

Properties of coefficients
In this section we shall consider properties of the Clebsch-Gordan coefficients m t i j in the decomposition (3.3) for a bialgebra H with decomposition (1.1) and with additional properties from Sect. 3. By Proposition 3.2 and Proposition 1.5 [1], the bimodule E g ⊗ M * i can be identified with M i where hxr = g, h · t r · x for all h, r ∈ H and x ∈ M i .
The bimodule M j ⊗ E * g can be identified with M i where hxr = hx g, r for all h, r ∈ H and x ∈ M i . Finally, the bimodule M i ⊗ M * j is identified with M i ⊗ M j where hxr = hx · t r for all h, r ∈ H and x ∈ M i .