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The Cubic Dirac Operator for Infinite-Dimensonal Lie Algebras

Published online by Cambridge University Press:  20 November 2018

Eckhard Meinrenken
Eckhard Meinrenken*
Affiliation:
University of Toronto, Department of Mathematics, 40 St. George Street, Toronto, ON M5S 2E4 email: [email protected]
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Abstract

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Let $\mathfrak{g}\,=\,{{\oplus }_{i\in \mathbb{Z}}}{{\mathfrak{g}}_{i}}$ be an infinite-dimensional graded Lie algebra, with dim ${{\mathfrak{g}}_{i}}\,<\,\infty $, equipped with a non-degenerate symmetric bilinear form $B$ of degree 0. The quantum Weil algebra ${{\hat{\mathcal{W}}}_{\mathfrak{g}}}$ is a completion of the tensor product of the enveloping and Clifford algebras of $\mathfrak{g}$. Provided that the Kac–Peterson class of $\mathfrak{g}$ vanishes, one can construct a cubic Dirac operator $\mathcal{D}\,\in \,\hat{\mathcal{W}}(\mathfrak{g})$, whose square is a quadratic Casimir element. We show that this condition holds for symmetrizable Kac– Moody algebras. Extending Kostant's arguments, one obtains generalized Weyl–Kac character formulas for suitable “equal rank” Lie subalgebras of Kac–Moody algebras. These extend the formulas of G. Landweber for affine Lie algebras.

Keywords

22E6515A66
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 6 , 01 December 2011 , pp. 1364 - 1387
Copyright
Copyright © Canadian Mathematical Society 2011

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