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On principal realization of modules for the affine Lie algebra $ A_1^{(1)}$ at the critical level


Authors: Dražen Adamović, Naihuan Jing and Kailash C. Misra
Journal: Trans. Amer. Math. Soc. 369 (2017), 5113-5136
MSC (2010): Primary 17B69; Secondary 17B67, 17B68, 81R10
DOI: https://doi.org/10.1090/tran/7009
Published electronically: March 1, 2017
MathSciNet review: 3632562
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Abstract: We present complete realization of irreducible $ A_1^{(1)}$-modules at the critical level in the principal gradation. Our construction uses vertex algebraic techniques, the theory of twisted modules and representations of Lie conformal superalgebras. We also provide an alternative Z-algebra approach to this construction. All irreducible highest weight $ A_1^{(1)}$-modules at the critical level are realized on the vector space $ M_{\tfrac {1}{2}+\mathbb{Z}}(1)^{\otimes 2}$ where $ M_{\tfrac {1}{2} + \mathbb{Z}}(1) $ is the polynomial ring $ {\mathbb{C}}[\alpha (-1/2), \alpha (-3/2),\dots ]$. Explicit combinatorial bases for these modules are also given.


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Additional Information

Dražen Adamović
Affiliation: Department of Mathematics, University of Zagreb, Bijenička 30, 10 000 Zagreb, Croatia
Email: adamovic@math.hr

Naihuan Jing
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695
Email: jing@math.ncsu.edu

Kailash C. Misra
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695
Email: misra@math.ncsu.edu

DOI: https://doi.org/10.1090/tran/7009
Keywords: Vertex superalgebras, affine Lie algebras, Clifford algebras, Weyl algebra, lattice vertex operator algebras, critical level, Z-algebras
Received by editor(s): December 17, 2015
Published electronically: March 1, 2017
Article copyright: © Copyright 2017 American Mathematical Society