Hopf subalgebras and tensor powers of generalized permutation modules
- Autores: Lars Kadison
- Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 218, Nº 2, 2014, págs. 367-380
- Idioma: inglés
- DOI: 10.1016/j.jpaa.2013.06.008
- Enlaces
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Resumen
By means of a certain module V and its tensor powers in a finite tensor category, we study a question of whether the depth of a Hopf subalgebra R of a finite-dimensional Hopf algebra H is finite. The module V is the counit representation induced from R to H, which is then a generalized permutation module, as well as a module coalgebra. We show that if in the subalgebra pair either Hopf algebra has finite representation type, or V is either semisimple with R. pointed, projective, or its tensor powers satisfy a Burnside ring formula over a finite set of Hopf subalgebras including R, then the depth of R in H is finite. One assigns a nonnegative integer depth to V, or any other H-module, by comparing the truncated tensor algebras of V in a finite tensor category and so obtains upper and lower bounds for depth of a Hopf subalgebra. For example, a relative Hopf restricted module has depth 1, and a permutationmodule of a corefree subgroup has depth less than the number of values of its character.
