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Laplace operators and the $ \mathfrak{h}$ module structure of certain cohomology groups


Author: Floyd L. Williams
Journal: Trans. Amer. Math. Soc. 197 (1974), 1-57
MSC: Primary 22E45
DOI: https://doi.org/10.1090/S0002-9947-1974-0379761-5
MathSciNet review: 0379761
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Abstract: Let $ \mathfrak{n}$ be the maximal nilpotent ideal of a Borel subalgebra of a complex semisimple Lie algebra $ \mathfrak{g}$. Under the adjoint action $ \mathfrak{n},\mathfrak{g}/\mathfrak{n}$, and $ \mathfrak{n}'$ (the dual space of $ \mathfrak{n}$) are $ \mathfrak{n}$ modules. Laplace operators for these three modules are computed by techniques which extend those introduced by B. Kostant in [6]. The kernels of these operators are then determined and, in view of the existence of a Hodge decomposition, the detailed structure of the first degree cohomology groups of $ \mathfrak{n}$ with coefficients in $ \mathfrak{n},\mathfrak{g}/\mathfrak{n}$, and $ \mathfrak{n}'$ is obtained. These cohomology groups (spaces) are described, in fact, as completely reducible modules of a Cartan subalgebra $ \mathfrak{h}$ of $ \mathfrak{g}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0379761-5
Keywords: Complex semisimple Lie algebra, Lie algebra cohomology, coboundary operator, Laplace operator, irreducible representation, highest weight
Article copyright: © Copyright 1974 American Mathematical Society

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