Note on Laplace operators and homologies of a few Lie subalgebras of $A_1^{(1)}$

Abstract

We investigate a spectrum of positive self-adjoint operator $\Gk$ (Laplace operator) acting in the external complexes of some interesting subalgebras $\Lk$ of Lie algebra $A_1^{(1)}$. We obtain an explicit formula for the action of $\G_k$. This formula is used to compute the homology of $\Lk$ with trivial coefficients for $k=-1,0,1,2$. In these cases we show that a spectrum of $\G_k$ is the set of non-negative integers with a finite multiplicity of each eigenvalue for $k=-1,0$, infinite one for $k=1,2$. We also found the generating functions for the multiplicities of $\G_k$-eigenvalues (in appropriate sense for $\G_1$ and $\G_2$).