Algebraic quotient modules and subgroup depth

作者：Hernandez Alberto; Kadison Lars^*; Young Christopher

来源：ABHANDLUNGEN AUS DEM MATHEMATISCHEN SEMINAR DER UNIVERSITAT HAMBURG, 2014, 84(2): 267-283.

DOI：10.1007/s12188-014-0097-3

• Adjoint action
• Cartan matrix
• Conditionally faithful
• Core of subgroup
• Cross product
• Drinfeld double
• Green ring

In Kadison J Pure Appl Alg 218:367-380, (2014) 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 is equivalent to the -module coalgebra representing an algebraic element in the Green ring of or . 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 has finite depth in is equivalent to determining if has finite depth in its smash product . A necessary condition is obtained for finite depth from stabilization of a descending chain of annihilator ideals of tensor powers of . As an application of these topics to a centerless finite group , we prove that the minimum depth of its group -algebra in the Drinfeld double is an odd integer, which determines the least tensor power of the adjoint representation that is a faithful -module.

• 出版日期2014-10