Geometrically reductive Hopf algebras
type: Item
instance of: scholarly article
title: Geometrically reductive Hopf algebras
zbMATH Open document ID: 0803.16038
DOI: 10.1016/0021-8693(92)90088-4
author: Heloisa Borsari
author: Walter Ricardo Ferrer Santos
published in: Journal of Algebra
publication date: 1993-03-29T00:00:00Z
review text: The authors define the notion of geometric reductivity for arbitrary commutative Hopf algebras \(C\) as follows: for every comodule \(M\) and every comodule homomorphism \(\lambda \in M^*\) there is a symmetric power \(r\) such that \(S^ r(\lambda)\) does not vanish on \(S^ r(M)^ C\), the invariants of the \(r\)-th symmetric power of \(M\). The exponent of reductivity of a pair \((M,\lambda)\) is the minimal such \(r\). It is \(p^ n\) (in characteristic \(p\)) or 1. Properties of geometric reductivity under Hopf algebra maps are studied. Furthermore it is shown that every finitely generated commutative \(C\)-comodule algebra has a finitely generated algebra of invariants. This is applied to the dual of a restricted enveloping algebra of a (f.d.) restricted Lie algebra \(u^*({\mathcal L})\), which is geometrically reductive.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 130509
zbMATH Keywords: geometric reductivity
zbMATH Keywords: commutative Hopf algebras
zbMATH Keywords: symmetric power
zbMATH Keywords: exponent of reductivity
zbMATH Keywords: Hopf algebra maps
zbMATH Keywords: finitely generated commutative \(C\)-comodule algebra
zbMATH Keywords: finitely generated algebra of invariants
zbMATH Keywords: restricted enveloping algebra
reviewed by: Bodo Pareigis
MaRDI profile type: Publication
full work available at URL: 0021-8693(92)90088-4
OpenAlex ID: W4213308345
cites work: Cohomology of comodules
cites work: Q3717277
cites work: Reductive groups are geometrically reductive
cites work: Q5722707
cites work: Lectures on introduction to moduli problems and orbit spaces
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cites work: Q5590089
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description: scientific article; zbMATH DE number 130509