Leibniz Homology and the James Model

Abstract

In this paper we study the geometry behind an algebraic construction of LODAY known as Leibniz homology, HL,, [8] [9]. These homology groups provide a noncommutative setting for Lie algebra homology much like cyclic homology is a noncommutative setting for de Rham cohomology. For an algebra A over Q, CUVIER [3] and LODAY [8] have shown that HL,(gI(A)) is rationally isomorphic to a tensor algebra involving the Hochschild homology groups of A. To put this result in perspective, recall that LODAY and QUILLEN [lo] have shown that H,(gI(A)), the Lie algebra homology of gI(A), is isomorphic to the graded exterior algebra A ( H C , -(A)), where H C , denotes cyclic homology. The calculation of HL,(gI(A)) is a lifting of the Lie algebra homology calculation to a certain tensor algebra. For a connected space X , it is also known [6] that the singular homology of QZX is isomorphic to a tensor algebra on the reduced homology of X . Here s2 denotes the based loop space and C is the reduced suspension. The tensor algebra in both cases can be explained in terms of certain presimplicial structures on the underlying space or chain complex. When A is the group ring Q [ G ] for a discrete group G, we establish an isomorphism between HL,(gI(A)) and the singular homology of a certain space involving the loops suspension functor and BG, (2.14). Here BG denotes the classifying space of G. Furthermore, we recover the James model [6] on BG in terms of various presimplicial layers suggested by the calculation of HL,(gI(Q[G])). The relation between the James model and Leibniz homology is the main geometric idea of this paper.Through an application of invariant theory to the chain complex defining Leibniz homology [8], the symmetric groups appear early in this paper. We define a certain multi-presimplicial stratification of these groups in which the m-presimplicial elements are precisely those permutations which can be expressed as the product of m-many disjoint cycles. This algebraic construction allows us to give an alternate proof of the computation of HL,(gI(A)). When this stratification is shifted slightly in dimension, the result is a chain complex for QCBG which is valid even over the integers.

On the degenerations of solvable Leibniz algebras

“…The noncommutative analog of Lie algebras are Leibniz algebras, discovered by Loday when he handled periodicity phenomena in algebraic K-theory [15]. This algebraic structure found applications in several fields as Physics and Geometry [9,14,16,17].…”