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The Cubic Dirac Operator for Infinite-Dimensonal Lie Algebras

  Published:2011-06-25
 Printed: Dec 2011
  • Eckhard Meinrenken,
    University of Toronto, Department of Mathematics, 40 St. George Street, Toronto, ON M5S 2E4
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Abstract

Let $\mathfrak{g}=\bigoplus_{i\in\mathbb{Z}} \mathfrak{g}_i$ 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 $\widehat{\mathcal{W}}\mathfrak{g}$ 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 $\mathcal{D}\in\widehat{\mathcal{W}}(\mathfrak{g})$, 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.
MSC Classifications: 22E65, 15A66 show english descriptions Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05]
Clifford algebras, spinors
22E65 - Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05]
15A66 - Clifford algebras, spinors
 

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