关于拉普拉斯算子和$A_1^{(1)}$的几个李子代数的同调的注记
Note on Laplace operators and homologies of a few Lie subalgebras of $A_1^{(1)}$
摘要：
研究了正自伴算子$\Gk$（Laplace算子）作用在李代数$A_1^{(1)}$的几个有趣子代数$\Lk$的外复形上的谱.我们得到了$\G_k$作用的显式公式。该公式用于计算具有$k=-1,0,1,2$的平凡系数的$\Lk$的同调。在这些情况下，我们证明了$\G_k$的谱是一组非负整数，其中每个特征值的重数对于$k=-1,0$是有限的，对于$k=1,2$是无限的。我们还找到了$\G_k$ -本征值的多重性的生成函数（在适当的意义下，对于$\G_1$和$\G_2$）。
英文摘要：
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 $\G_2$).
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