On cocommutative Hopf algebras of finite representation type
type: Item
instance of: scholarly article
title: On cocommutative Hopf algebras of finite representation type
zbMATH Open document ID: 0981.16030
author: Rolf Farnsteiner
author: Detlef Voigt
published in: Advances in Mathematics
publication date: 2001-12-11T00:00:00Z
review text: Let \(\mathcal G\) be a finite algebraic group over an algebraically closed field \(k\) of positive characteristic. The structure of the ``infinitesimal part'' (i.e. the largest normal infinitesimal subgroup \({\mathcal G}^0\subset{\mathcal G}\)) is determined up to the classification of what are called ``\(V\)-uniserial groups''. A group \(\mathcal U\) is called \(V\)-uniserial if the Verschiebung \(V_{\mathcal U}\) induces an exact sequence \({\mathcal U}^{(p)}\to{\mathcal U}\to{_p\alpha_k}\to e_k\). The term \(V\)-uniserial is used because in this case \(\mathcal U\) has exactly one composition series as its subgroups are of the form \(V_{\mathcal U}^i({\mathcal U}^{(p^i)})\). Initially, the paper focuses on infinitesimal groups of finite representation type. It is shown that such a group is supersolvable, and this observation leads to a description of the structure of such groups. The principal blocks are representation-finite, as is shown in Theorem 2.7. The classification is shown to reduce to the classification of \(V\)-uniserial groups, however the classification of these groups does not appear here -- the authors state that its solution ``involves different techniques and is therefore relegated to a separate paper.'' Indeed, the calculations here are homological, whereas it is stated that future papers will rely more on geometry. The topic is then generalized to dimension zero algebraic groups that are representation-finite. It is shown that such a Hopf algebra is representation-finite if and only if the constant and infinitesimal components are representation-finite and at least one of the components is semi-simple. Finally, it is shown how some of the results are related to work by \textit{A. Premet} [Math. Z. 228, No. 2, 255-282 (1998; Zbl 0913.17011)].
reviewed by: Alan Koch
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1545684
zbMATH Keywords: cocommutative Hopf algebras
zbMATH Keywords: finite algebraic groups
zbMATH Keywords: infinitesimal subgroups
zbMATH Keywords: uniserial groups
zbMATH Keywords: finite representation type
MaRDI profile type: Publication
OpenAlex ID: W2069281654
cites work: Q4318498
cites work: Q3999421
cites work: Q3997902
cites work: Q3999894
cites work: The varieties and the cohomology ring of a module
cites work: Q5602064
cites work: Blocks of tame representation type and related algebras
cites work: Q4883287
cites work: Periodicity and representation type of modular Lie algebras.
cites work: Restricted Lie algebras with bounded cohomology and related classes of algebras
cites work: Support varieties for restricted Lie algebras
cites work: Geometry of p-unipotent Lie algebras
cites work: Modular Representation Theory of Lie Algebras
cites work: Cohomology of finite group schemes over a field
cites work: Indecomposable Representations and the Loop-Space Operation
cites work: Indecomposable representations at characteristic \(p\)
cites work: Q3998721
cites work: Homological characteristics of pro-uniserial rings
cites work: The Representation Type of Algebras and Subalgebras
cites work: Q4087381
cites work: An Associative Orthogonal Bilinear Form for Hopf Algebras
cites work: A Hopf Algebra Freeness Theorem
cites work: Über Untergruppen endlicher algebraischer Gruppen
cites work: Q3822398
cites work: Restricted Lie algebras of bounded type
cites work: Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture
cites work: Complexity of Lie algebra representations and nilpotent elements of the stabilizers of linear forms
cites work: The Representation Type of Self-Injective Algebras
cites work: Q4039778
cites work: Infinitesimal 1-parameter subgroups and cohomology
cites work: Support varieties for infinitesimal group schemes
cites work: Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen
cites work: Q3971060
cites work: Q3884236
DOI: 10.1006/AIMA.2000.1920
description: scientific article; zbMATH DE number 1545684