# Lie algebra cohomology and generating functions

## Alexei Tolpygo

Let $\fg$ be a simple Lie algebra, $V$ an irreducible $\fg$-module, $W$ the Weyl group and $\fb$ the Borel subalgebra of $\fg$, $\fn = [\fb, \fb ]$, $\fh$ the Cartan subalgebra of $\fg$. The Borel-Weil-Bott theorem states that the dimension of $H^{i}(\fn; V)$ is equal to the cardinality of the set of elements of length $i$ from $W$. Here a more detailed description of $H^{i}(\fn; V)$ as an $\fh$-module is given in terms of generating functions.

Results of Leger and Luks and Williams who described $H^{i}(\fn; \fn)$ for $i\leq 2$ are generalized: $\dim H^{*}(\fn; \Lambda^{*}(\fn))$ and $\dim H^{i}(\fn; \fn)$ for $i\leq 3$ are calculated and $\dim H^{i}(\fn; \fn)$ as function of $i$ and rank $\fg$ is described for the calssical series.

Homology, Homotopy and Applications, Vol. 6(2004), No. 1, pp. 59-85

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