## Homology, Homotopy and Applications

### Lie algebra cohomology and generating functions

Alexei Tolpygo

#### Abstract

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

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

#### Article information

Source
Homology Homotopy Appl., Volume 6, Number 1 (2004), 59-85.

Dates
First available in Project Euclid: 13 February 2006

https://projecteuclid.org/euclid.hha/1139839545

Mathematical Reviews number (MathSciNet)
MR2061568

Zentralblatt MATH identifier
1102.17005

Subjects
Primary: 17B56: Cohomology of Lie (super)algebras

#### Citation

Tolpygo, Alexei. Lie algebra cohomology and generating functions. Homology Homotopy Appl. 6 (2004), no. 1, 59--85. https://projecteuclid.org/euclid.hha/1139839545