Homology of the Lie algebra corresponding to a poset

In this thesis, we study the spectral resolution of the Laplacian {dollar}{lcub}cal L{rcub}{dollar} of the Koszul complex of the Lie algebras corresponding to a certain class of posets.; Given a poset P on the set {dollar}{lcub}1,2,...,{lcub}rm n{rcub}{rcub},{dollar} we define the nilpotent Lie algebra {dollar}Lsb{lcub}P{rcub}{dollar} to be the span of all elementary matrices {dollar}zsb{lcub}x,y{rcub}{dollar}, such that x is less than y in P. In this thesis, we will make a decisive step toward calculating the Lie algebra homology of {dollar}Lsb{lcub}P{rcub}{dollar} in the case that the Hasse diagram of P is a rooted tree.; We show that the Laplacian {dollar}{lcub}cal L{rcub}{dollar} is significantly simplified when the posets considered are those whose Hasse diagram is a tree. The main result of this thesis determines the spectral resolutions of three commuting linear operators whose sum is the Laplacian {dollar}{lcub}cal L{rcub}{dollar} of the Koszul complex of {dollar}Lsb{lcub}P{rcub}{dollar} in the case that the Hasse diagram is a rooted tree.; We show that these eigenvalues are integers, give a combinatorial indexing of these eigenvalues and describe the corresponding eigenspaces in representation-theoretic terms. The homology of {dollar}Lsb{lcub}P{rcub}{dollar} is represented by the nullspace of {dollar}{lcub}cal L{rcub}{dollar}, so in future work, these results should allow for the homology to be effectively computed.; These results have several interesting corollaries that are of a combinatorial nature. We will state one. Let P be a rooted tree on n nodes and let {dollar}Sigma{dollar} be the sum in the group algebra of {dollar}Ssb{lcub}n{rcub}{dollar} of all transpositions (i, j) such that i is on the unique path from j to the root in P. Then {dollar}Sigma{dollar} acting on C{dollar}Ssb{lcub}n{rcub}{dollar} by left multiplication has non-negative integer eigenvalues and the corresponding eigenspaces can be identified in representation-theoretic terms.