Mathematics – Representation Theory
Scientific paper
2011-06-15
Mathematics
Representation Theory
33 pages
Scientific paper
Recently, Kashina, Montgomery and Ng introduced the $n$-th indicator $\nu_n(H)$ of a finite-dimensional Hopf algebra $H$ and showed that it has several interesting properties such as the gauge invariance. The aim of this paper is to investigate properties of indicators finite-dimensional Hopf algebras, especially those of non-semisimple one. We develop some techniques to compute indicators and apply them to study relations between indicators and the quasi-exponent. Also relations between indicators and the length of the coradical filtration are discussed. Our results are also applied to the finite-dimensional pointed Hopf algebra $u(\mathcal{D}, \lambda, \mu)$ introduced by Andruskiewitsch and Schneider. The Taft algebra and the small quantum group associated with $\mathfrak{sl}_2$ are examples of $u(\mathcal{D}, \lambda, \mu)$. It turns out that indicators of them can be expressed by special values of the generating function of certain type of partitions at roots of unity.
Shimizu Kenichi
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