On the adjoint representation of a hopf algebra. Issue 4 (11th November 2020)

• Record Type: Journal Article
• Authors: Kolb, Stefan; Lorenz, Martin; Nguyen, Bach; Yammine, Ramy
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 63:Issue 4(2020)
• Page Start: 1092
• Page End: 1099
• Publication Date: 2020-11-11
• Subjects: Infinite-dimensional Hopf algebra, -- cocommutative Hopf algebra, -- pointed Hopf algebra, -- Hopf subalgebra, -- adjoint representation, -- locally finite part, -- tensor functor, -- coideal subalgebra, -- Δ-methods, -- Dietzmann's Lemma; 16T05, -- 16T2; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091520000358
• Languages: English
• ISSNs: 0013-0915

Abstract

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.