Cohomology of Oriented Tree Diagram Lie Algebras

TL;DR

This paper studies the cohomology of generalized oriented tree diagram Lie algebras using Hodge Laplacian, proving conjectures and deriving generating functions for Betti numbers, extending classical Lie algebra results.

Contribution

It generalizes the cohomological analysis of tree diagram Lie algebras, proves the total rank and b2 conjectures, and derives generating functions and identities extending Bott's classical results.

Findings

Proved the total rank conjecture for these Lie algebras.

Established the b2-conjecture for the class of algebras.

Derived generating functions for Betti numbers using Young tableaux.

Abstract

Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this paper, we use Hodge Laplacian to study the cohomology of these Lie algebras. The "total rank conjecture" and "$b_2$-conjecture" for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler-Poincare principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Bott's classical result in the case of special linear Lie algebras.