On a Plus-Construction for Algebras over an Operad

Abstract

We prove a theorem analogous to Quillen's plus-construction in the category of algebras over an operad. For that purpose we prove that this category is a closed model category and prove the existence of an obstruction theory. We apply further this plus-construction for the specific cases of Lie algebras and Leibniz algebras which are a noncommutative version of Lie algebras: let sl.A/ be the kernel of the trace map gl.A/! A=(A;A), where A is an associative algebra with unit and gl.A/ is the Lie algebra of matrices over A. Then the homotopy of sl.A/ C in the category of Lie algebras is the cyclic homology of A whereas it is the Hochschild homology ofA in the category of Leibniz algebras. Mathematics Subject Classifications ( 1991): 18Dxx, 17A30, 55S35.