Some Hopf algebras of dimension $72$ without the Chevalley property
Abstract
In this paper, we consider the Drinfeld double $\D$ of a $12$dimensional Hopf algebra $\C$ over an algebraically closed field of characteristic zero whose coradical is not a subalgebra and describe its simple modules, projective covers of the simple modules and show that it is of wild representation type. Moreover, we show that the Nichols algebras associated to nonsimple indecomposable modules are infinitedimensional. In particular, for any object $V$ in $\CYD$, if $\BN(V)$ is finitedimensional, then $V$ must be semisimple. Finally, we describe the Nichols algebras associated to partial simple modules in terms of generators and relations. As a byproduct, we obtain some Hopf algebras of dimension $72$ without the Chevalley property, that is, the coradical is not a subalgebra.
 Publication:

arXiv eprints
 Pub Date:
 December 2016
 DOI:
 10.48550/arXiv.1612.04987
 arXiv:
 arXiv:1612.04987
 Bibcode:
 2016arXiv161204987H
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Representation Theory
 EPrint:
 37 pages