Some bismash products that are not group algebras

Abstract

We show that an infinite family of Hopf algebras that arise as a subfamily of the so-called bismash products constructed from factorisations of symmetric groups do not have the structure (as algebras) of group algebras.

Representations of Some Hopf Algebras Associated to the Symmetric Group S n

“…Then the value of the Frobenius-Schur indicator ofV is exactly the same as the indicator for V as a simple kF-module. (2) Assume that x 2 = 1. Then the value of the Frobenius-Schur indicator ofV is 0.…”

“…First, any bicrossed product constructed from the factorization S n = S n−1 C n = C n S n−1 gives nothing new; that is, it is isomorphic to H n or J n [17]; see Example 2.4. Second, it has recently been shown by Collins that the algebra structure of H n is unlike that of any group algebra; that is, there does not exist any finite group G such that the set of degrees of the irreducible representations of G is the same as the set of degrees of the irreducible representations of H n , for any prime p > 3 and n = p, p + 1, or p + 2 [2].…”

“…Now y −1 = (y −1 a) −1 ⇐⇒ y −1 = y (y −1 a) by Lemma 4.2, and so[2] becomes[2] = 1 |F| y∈G a∈F y −1 ,y p y #(y −1 a)a. (4.1)Now consider an irreducible H-moduleV with characterχ.…”

On Quasitriangular Structures in Hopf Algebras Arising from Exact Group Factorizations

“…Remark 4.3. We have shown, as part of the proof of Proposition 3.2 and Theorem 4.2, that the bicrossed product Hopf algebra H is not twist equivalent to the group algebra of G. For the case where G = S n , n = p + 1 or p + 2, p > 3 a prime number, and H corresponds to the exact factorization considered in Section 3, this fact follows from the main result of [5] that says that H is not isomorphic as an algebra to any group algebra. The analogous result is true for a bismash product (split extension) Hopf algebra associated to the groups PGL 2 (q), q = 2, 3, as shown in [4].…”

On the algebra structure of some bismash products

“…Hopf algebras H ′ obtained from such a H by a comultiplication twist certainly have the same algebra structure as H. Hence, information obtained about the algebra structure of H applies also to this wider class of examples. It was shown in [1] that an infinite family of bismash products constructed via factorizations of symmetric groups do not have the structure, as algebras, of group algebras. In the present paper we prove a similar result corresponding to factorizations of P GL 2 (q).…”