On coalgebras whose complex is a quadratic algebra
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.
quotient complexes with quadratic relations
scientific article; zbMATH DE number 563461
scientific article