Algebra depth in tensor categories

作者：Kadison Lars^*

来源：Bulletin of the Belgian Mathematical Society Simon Stevin, 2016, 23(5): 721-752.

DOI：10.36045/bbms/1483671623

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

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.

• 出版日期2016