Level one Weyl modules for toroidal Lie algebras
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Level one Weyl modules for toroidal Lie algebras Ryosuke Kodera1 Received: 8 January 2020 / Revised: 8 January 2020 / Accepted: 29 July 2020 © Springer Nature B.V. 2020
Abstract We identify level one global Weyl modules for toroidal Lie algebras with certain twists of modules constructed by Moody–Eswara Rao–Yokonuma via vertex operators for type ADE and by Iohara–Saito–Wakimoto and Eswara Rao for general type. The twist is given by an action of SL2 (Z) on the toroidal Lie algebra. As a by-product, we obtain a formula for the character of the level one local Weyl module over the toroidal Lie algebra and that for the graded character of the level one graded local Weyl module over an affine analog of the current Lie algebra. Keywords Toroidal Lie algebra · Weyl module · Character · Vertex operator Mathematics Subject Classification Primary 17B67; Secondary 17B10 · 17B65 · 17B69
1 Introduction 1.1 Motivation We study global/local Weyl modules for toroidal Lie algebras and an affine analog of current Lie algebras. The notion of Weyl modules for affine Lie algebras has been introduced by Chari–Pressley in [5] as a family of integrable highest weight modules with a universal property. Later Chari–Loktev initiated in [4] to study Weyl modules for current Lie algebras in a graded setting. The graded characters of local Weyl modules for current Lie algebras have been studied by many authors. Now they are known to coincide with Macdonald polynomials specialized at t = 0, a.k.a. q-Whittaker functions (Chari–Loktev [4], Fourier–Littelmann [10], Naoi [17], Sanderson [19], Ion [12], and Lenart–Naito–Sagaki–Schilling–Shimozono [14]).
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Ryosuke Kodera [email protected] http://www2.kobe-u.ac.jp/∼kryosuke/ Department of Mathematics, Graduate School of Science, Kobe University, Kobe, Japan
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R. Kodera
Toroidal Lie algebras are natural generalization of affine Lie algebras. For a finitedimensional simple Lie algebra g, the corresponding toroidal Lie algebra gtor is defined as the universal central extension of the double loop Lie algebra g ⊗ C[s ±1 , t ±1 ] with the degree operators. We can also consider a Lie algebra g+ tor which is defined by replacing C[s ±1 , t ±1 ] with C[s, t ±1 ]. See Sect. 2.2 for precise definitions. We expect that the characters of Weyl modules for gtor and g+ tor produce a very interesting class of special functions. In this article, we study the first nontrivial example: the Weyl module associated with the level one dominant integral weight. A big difference between the toroidal and the affine Lie algebra is the structure of their centers. The toroidal Lie algebra without the degree operators has an infinitedimensional center, while the center of the affine Lie algebra is one-dimensional. The Weyl modules are examples of modules over the toroidal Lie algebra on which the action of the center does not factor a finite-dimensional quotient. We note that Chari– Le have studied in [3] local Weyl modules for a quotient of the toroidal Lie algebra. The resulting quotient is an extension of the d
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