On coalgebras whose complex is a quadratic algebra
type: Item
instance of: scholarly article
title: On coalgebras whose complex is a quadratic algebra
zbMATH Open document ID: 0861.16024
DOI: 10.1016/0034-4877(93)90029-E
author: Jerzy Różański
published in: Reports on Mathematical Physics
publication date: 1994-06-16T00:00:00Z
review text: The author presents an approach to the cohomology theory of coalgebras via quotient complexes with quadratic relations. Two types of coalgebras, coassociative and Lie coalgebras, are special cases of this theory. First, the cohomology theory for coassociative coalgebras is recalled, which is very well known as a variant of the Hochschild cohomology. Similarly as for the Hochschild cohomology, the cohomology of Lie algebras is determined by conditions for extension of comultiplication to the derivation of the exterior algebra. The author describes a method of determining cohomologies of coalgebras by introducing equations for quadratic relations \(R\subseteq L\otimes L\) in the tensor algebra \(TL\) where \(L\) is a \(k\)-vector space over a field \(k\). Solutions to the equations for cohomology theory of quotient complexes are possible when the dimension of \(I^3=R\otimes L+L\otimes R\) is known. The number of equations is determined by this dimension. Calculation of this problem is equivalent to the calculation of the third term in the Poincaré series for the quotient tensor algebra \((TL)/I\). The approach of this paper could have a possible application to the theory of deformations of algebras.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16S80
Mathematics Subject Classification ID: 17B56
zbMATH DE Number: 563461
zbMATH Keywords: quotient complexes with quadratic relations
zbMATH Keywords: coassociative coalgebras
zbMATH Keywords: Hochschild cohomology
zbMATH Keywords: cohomology of Lie algebras
zbMATH Keywords: deformations of algebras
reviewed by: Fang Li
MaRDI profile type: Publication
cites work: Cohomology Theory of Lie Groups and Lie Algebras
cites work: Q5788192
cites work: Quantum symmetry
cites work: On the cohomology groups of an associative algebra
cites work: Lie coalgebras
cites work: Skew derivations and \(U_ q(sl(2))\)
full work available at URL: 0034-4877(93)90029-e
OpenAlex ID: W2002801644
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description: scientific article; zbMATH DE number 563461