2019/05/28 by Stefan Kolb, Martin Lorenz, Bach Nguyen +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Functor #Hopf algebra #Lemma (botany) #Quantum group #Quasitriangular Hopf algebra #Representation theory of Hopf algebras #Tensor algebra #math.RT #msc:16T05 #msc:16T20
paper · pdf · doi:10.1017/s0013091520000358
published as Proceedings of the Edinburgh Mathematical Society 63 (2020) 1092-1099 · This updates an earlier version, with improved results, additional references, and a new co-author (Stefan Kolb)
openalex created_date 2019/05/16 · arxiv created 2019/05/28 · openalex publication_date 2020/11/01 · arxiv updated 2021/01/13 · openalex updated_date 2026/08/05
Abstract We consider the adjoint representation of a Hopf algebra H focusing on the locally finite part, H_\textrm ad fin , defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative H (i.e., H is finitely generated as module over a cocommutative Hopf subalgebra), we show that H_\textrm ad fin is a Hopf subalgebra of H . This is a consequence of the fact, proved here, that locally finite parts yield a tensor functor on the module category of any virtually pointed Hopf algebra. For general Hopf algebras, H_\textrm ad fin is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.