Representations of the Lie algebra of vector fields on a torus and the chiral de Rham complex
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- by Yuly Billig and Vyacheslav Futorny PDF
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Abstract:
The goal of this paper is to study the representation theory of a classical infinite-dimensional Lie algebra – the Lie algebra $\textrm {Vect}{\mathbb {T}}^N$ of vector fields on an $N$-dimensional torus for $N > 1$. The case $N=1$ gives a famous Virasoro algebra (or its centerless version - the Witt algebra). The algebra $\textrm {Vect}{\mathbb {T}}^N$ has a natural class of tensor modules parametrized by finite-dimensional modules of $gl_N$. Tensor modules can be used in turn to construct bounded irreducible modules for $\textrm {Vect}{\mathbb {T}}^{N+1}$ (induced from $\textrm {Vect}{\mathbb {T}}^N$), which are the focus of our study. We solve two problems regarding these bounded modules: we construct their free field realizations and determine their characters. To solve these problems we analyze the structure of the irreducible $\Omega ^1 \left ( \mathbb {T}^{N+1} \right ) / d \Omega ^0 \left ( \mathbb {T}^{N+1} \right ) \rtimes \textrm {Vect}{\mathbb {T}}^{N+1}$-modules constructed in a paper by the first author. These modules remain irreducible when restricted to the subalgebra $\textrm {Vect}{\mathbb {T}}^{N+1}$, unless they belong to the chiral de Rham complex, introduced by Malikov-Schechtman-Vaintrob (1999).References
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Additional Information
- Yuly Billig
- Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
- Email: billig@math.carleton.ca
- Vyacheslav Futorny
- Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 05315-970 Brasil
- MR Author ID: 238132
- Email: futorny@ime.usp.br
- Received by editor(s): August 31, 2011
- Received by editor(s) in revised form: September 11, 2012
- Published electronically: April 25, 2014
- Additional Notes: The first author was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada
The second author was supported in part by the CNPq grant (301743/2007-0) and by the Fapesp grant (2010/50347-9) - © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 4697-4731
- MSC (2010): Primary 17B66, 17B67; Secondary 17B69
- DOI: https://doi.org/10.1090/S0002-9947-2014-05959-X
- MathSciNet review: 3217697
Dedicated: Dedicated to Yuri Alexandrovich Bahturin