On Some Families of Modules for the Current Algebra

Abstract

Given a finite-dimensional module V for a finite-dimensional, complex semi-simple Lie algebra \(\mathcal {g}\), and a positive integer m, we construct a family of graded modules for the current algebra \(\mathcal {g}[t]\) indexed by simple C \(\mathcal {S}_{m}\)-modules. These modules are free of finite rank for the ring of symmetric polynomials and so can be localized to give finite-dimensional graded \(\mathcal {g}[t]\)-modules. We determine the graded characters of these modules and show that these graded characters admit a curious duality.

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Correspondence to Matthew Bennet.

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Presented by Vyjayanthi Chari.

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Bennet, M., Jenkins, R. On Some Families of Modules for the Current Algebra. Algebr Represent Theor 20, 197–208 (2017). https://doi.org/10.1007/s10468-016-9637-0

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Keywords

  • Current algebra
  • Lie algebra
  • Tilting module
  • Symmetric group
  • Graded module