Algebraic quotient modules and subgroup depth

Abstract

Abstract. In [28] it was shown that subgroup depth may be computed from the permutation module of the left or right cosets: this holds more generally for a Hopf subalgebra, from which we note in this paper that finite depth of a Hopf subalgebra R ⊆ H is equivalent to the H-module coalgebra Q = H/R + H representing an algebraic element in the Green ring of H or R. This approach shows that subgroup depth and the subgroup depth of the corefree quotient lie in the same closed interval of length one. We also establish a previous claim that the problem of determining if R has finite depth in H is equivalent to determining if H has finite depth in its smash product Q * #H. A necessary condition is obtained for finite depth from stabilization of a descending chain of annihilator ideals of tensor powers of Q. As an application of these topics to a centerless finite group G, we prove that the minimum depth of its group C -algebra in the Drinfeld double D(G) is an odd integer, which determines the least tensor power of the adjoint representation Q that is a faithful C G-module.

Algebra depth in tensor categories

“…9]. The question in general is only interesting for the projective-free summands of Q, since projectives form a finite rank ideal in the representation ring [15]. If either R or H has finite representation type (e.g., is semisimple, Nakayama serial), Q is similarly algebraic.…”

“…for all c ∈ C, h ∈ H. The depth d(A, M H ) and d(C, M H ) is a linear rescaling of the minimum depth of any object in M H defined in [31,15,16], not an important difference, though slightly more convenient in formulas given below. (1) .…”

A quantum subgroup depth

“…Drinfeld's quantum double D(H) of a Hopf algebra H is frequently applied to a group algebra. 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).…”

“…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].…”

An In-Depth Look at Quotient Modules

“…Conversely, if I = H, and Ann Q ⊗ℓQ = 0, then τ (Q ⊗ℓQ ) = H, which shows that L Q ≤ ℓ Q . Recall from[22] that an H-module W is conditionally faithful, if one of its tensor powers is faithful.…”