Frobenius algebras of corepresentations and group-graded vector spaces

Abstract

We consider Frobenius algebras in the monoidal category of right comodules over a Hopf algebra H. If H is a group Hopf algebra, we study a more general Frobenius type property and uncover the structure of graded Frobenius algebras. Graded symmetric algebras are also investigated.

Symmetric algebras in categories of corepresentations and smash products

“…Moreover, A * ∈ M H A , with the usual right A-action; this means that the A-module structure of A * is right H-colinear. It is known (see [6,Theorem 2.4]) that the following are equivalent: On the other hand, A * is also a left A-module in a natural way, but in general A * is not an object of A M H (with a similar compatibility condition for the A-action and H-coaction). However, A * is an object in A (S 2 ) M H , where A (S 2 ) is just the algebra A, with the H-coaction shifted by S 2 , i.e.…”

“…It is easy to see that A is (H, ε)-symmetric, i.e. graded symmetric in the terminology of [6], for example by using the results of [7], where the question whether any graded division algebra is graded symmetric is addressed. On the other hand, A is not (H, u)-symmetric.…”

“…In this example we use the additive notation for the operation of C 2 . Examples of such A are given in [6,Section 6]; the trivial extension associated to a finite dimensional algebra is one such example. Let µ : A → k be a linear map whose kernel does not contain non-zero left ideals of A.…”

Graded semisimple algebras are symmetric

“…It seems to be an interesting problem to understand the structure of Frobenius (symmetric) algebras in certain monoidal categories. A study in this direction was done in [6] for the category of vector spaces graded by an arbitrary group G. A finite dimensional G-graded algebra A = ⊕A g is symmetric in this category (we shortly say that A is graded symmetric), if A and A * are isomorphic as G-graded A, A-bimodules. Here the grading on A * is given by (A * ) g = {f ∈ A * | f (A h ) = 0 for any h = g −1 }.…”

“…For notation and terminology about graded algebras we refer to [18]. We recall from [6] that a finite dimensional G-graded algebra A is graded symmetric if and only if there exists a linear map λ : A → k such that λ(ab) = λ(ba) for any a, b ∈ A, λ(A g ) = 0 for any g = e, and Ker λ does not contain non-zero graded left ideals.…”