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

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.

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