The adjoint representation of group algebras and enveloping algebras

Abstract

In this paper we study the Hopf adjoint action of group algebras and enveloping algebras . We are particularly concerned with determining when these representations are faithful . Delta methods allow us to reclute the problerri to certain better behaved subalgebras . Nevertheless ; the problem remairis open in the finite group and finite-dirnensional Lie algebra cases .

A quantum subgroup depth

“…For example, in [27] it is proved that the adjoint representation of the symmetric group on itself is faithful, which rephrased in the terms of Q in this paper, shows that Q and Q ⊗ (2) have the same indecomposable constituents, or Q G has "depth 1" for the group C-algebra of a symmetric group on 3 or more letters (in its Drinfeld double with quotient module Q). It is also noted in [27,Lemma 1.3] that for G set equal to certain semidirect products of elementary abelian p-and q-groups, where p, q are primes such that p | q − 1, the adjoint action of G on itself is not faithful: from this recipe, a group G of order 108 with Q of depth 2 in D(G) is constructed in [17,Example 6.5]. For any semisimple Hopf subalgebra pair R ⊆ H, the depth of Q coincides with the length n of the chain of annihilator ideals of tensor powers of Q, i.e., the least n for which Ann Q ⊗(n) is a Hopf ideal [25,Theorem 3.14], [19].…”

“…The depth of a finite group G in its double D(G) (over C) is studied in [17], where it is shown to be closely related to the tensor power of the adjoint representation of G on itself at which it is faithful, a topic introduced and explored in [27]. For example, in [27] it is proved that the adjoint representation of the symmetric group on itself is faithful, which rephrased in the terms of Q in this paper, shows that Q and Q ⊗ (2) have the same indecomposable constituents, or Q G has "depth 1" for the group C-algebra of a symmetric group on 3 or more letters (in its Drinfeld double with quotient module Q). It is also noted in [27,Lemma 1.3] that for G set equal to certain semidirect products of elementary abelian p-and q-groups, where p, q are primes such that p | q − 1, the adjoint action of G on itself is not faithful: from this recipe, a group G of order 108 with Q of depth 2 in D(G) is constructed in [17,Example 6.5].…”

On some representations of the Drinfel'd double

“…In the same paper [18] it was proved that the adjoint action on S n is faithful, therefore ∼ is an equivalence relation and in this case there is only one equivalence class. (D(H )).…”

“…Therefore ω(C G (g)) ⊇ p i=0 ω(Q i ) = 0. In the same paper [18] it is shown that Ann(H ad ) = g∈G ω(C G (g)). Therefore Ann(H ad ) = 0 although Z(G) = 1.…”

“…Example 1 [18]. Let p and q be two prime numbers with q − 1 divisible by p. We will construct a group G such that Z(G) = 1 but Ann((kG) ad ) = 0.…”