Bulletin of the Belgian Mathematical Society - Simon Stevin

Algebra depth in tensor categories

Lars Kadison

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Study of the quotient module of a finite-dimensional Hopf subalgebra pair in order to compute its depth yields a relative Maschke Theorem, in which semisimple extension is characterized as being separable, and is therefore an ordinary Frobenius extension. We study the core Hopf ideal of a Hopf subalgebra, noting that the length of the annihilator chain of tensor powers of the quotient module is linearly related to the depth, if the Hopf algebra is semisimple. A tensor categorical definition of depth is introduced, and a summary from this new point of view of previous results are included. It is shown in a last section that the depth, Bratteli diagram and relative cyclic homology of algebra extensions are Morita invariants.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 23, Number 5 (2016), 721-752.

First available in Project Euclid: 6 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16D20: Bimodules 16D90: Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality 16T05: Hopf algebras and their applications [See also 16S40, 57T05] 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 20C05: Group rings of finite groups and their modules [See also 16S34]

subgroup depth Morita equivalent ring extensions Frobenius extension semisimple extension tensor category core Hopf ideals relative Maschke theorem


Kadison, Lars. Algebra depth in tensor categories. Bull. Belg. Math. Soc. Simon Stevin 23 (2016), no. 5, 721--752. doi:10.36045/bbms/1483671623. https://projecteuclid.org/euclid.bbms/1483671623

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