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.

Abstract. Danz computes the depth of certain twisted group algebra extensions in [11], which are less than the values of the depths of the corresponding untwisted group algebra extensions in [8]. In this paper, we show that the closely related h-depth of any group crossed product algebra extension is less than or equal to the h-depth of the corresponding (finite rank) group algebra extension. A convenient theoretical underpinning to do so is provided by the entwining structure of a right H-comodule algebra A and a right H-module coalgebra C for a Hopf algebra H. Then A ⊗ C is an A-coring, where corings have a notion of depth extending hdepth. This coring is Galois in certain cases where C is the quotient module Q of a coideal subalgebra R ⊆ H. We note that this applies for the group crossed product algebra extension, so that the depth of this Galois coring is less than the h-depth of H in G. Along the way, we show that subgroup depth behaves exactly like combinatorial depth with respect to the core of a subgroup, and extend results in [23] to coideal subalgebras of finite dimension.

The Green ring of the half quantum group H = U n (q) is computed in [9]. The tensor product formulas between indecomposables may be used for a generalized subgroup depth computation in the setting of quantum groups -to compute depth of the Hopf subalgebra H in its Drinfeld double D(H). In this paper the Hopf subalgebra quotient module Q (a generalization of the permutation module of cosets for a group extension) is computed and, as H-modules, Q and its second tensor power are decomposed into a direct sum of indecomposables. We note that the least power n, referred to as depth, for which Q ⊗(n) has the same indecomposable constituents as Q ⊗(n+1) is n = 2, since Q ⊗(2) contains all H-module indecomposables, which determines the minimum even depth d ev (H, D(H)) = 6. √ 8n+1−1 2 + 1, as computed in 1991 Mathematics Subject Classification. 16D20, 16D90.

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