Leibniz homology of extended Lie algebras
Let \(A\) be a commutative algebra, \({\mathfrak g}\) be a Lie algebra, and consider the extended Lie algebra \(A\otimes{\mathfrak g}\) with bracket \([a_1\otimes x_1,a_2\otimes x_2]= a_1a_2\otimes [x_1,x_2]\). For \({\mathfrak g}\) perfect the Leibniz homology \(HL_* \otimes{\mathfrak g}\) in dimensions two and three are computed in terms of the Hochschild homology of \(A\) and the coinvariants of the symmetric powers of \({\mathfrak g}\). Leibniz homology is a noncommutative analogue of Lie algebra homology which was introduced by Loday six years ago. The results here parallel those of \textit{A. Haddi} [Homologie de degré trois des algèbre de Lie étendues à une algèbre commutative, C. R. Acad. Sci., Paris, Sér. I 321, 965-968 (1995; Zbl 0846.17024)] obtained for the usual Lie algebra cohomology of \(A\otimes{\mathfrak g}\) where the Hochschild homology of \(A\) is replaced by its cyclic homology.
scientific article; zbMATH DE number 1132930
scientific article