Weyl modules for the twisted loop algebras (original) (raw)
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Global Weyl modules for the twisted loop algebra
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 2013
We define global Weyl modules for twisted loop algebras and analyze their highest weight spaces, which are in fact isomorphic to Laurent polynomial rings in finitely many variables. We are able to show that the global Weyl module is a free module of finite rank over these rings. Furthermore we prove, that there exist injective maps from the global Weyl modules for twisted loop algebras into a direct sum of global Weyl modules for untwisted loop algebras. Relations between local Weyl modules for twisted and untwisted generalized current algebras are known; we provide for the first time a relation on global Weyl modules.
The block decomposition of finite-dimensional representations of twisted loop algebras
Pacific Journal of Mathematics, 2009
Let L σ (g) be the twisted loop algebra of a simple complex Lie algebra g with nontrivial diagram automorphism σ. Although the category Ᏺ σ of finite-dimensional representations of L σ (g) is not semisimple, it can be written as a sum of indecomposable subcategories (the blocks of the category). To describe these summands, we introduce the twisted spectral characters for L σ (g). These are certain equivalence classes of the spectral characters defined by Chari and Moura for an untwisted loop algebra L(g), which were used to provide a description of the blocks of finite-dimensional representations of L(g). Here we adapt this decomposition to parametrize and describe the blocks of Ᏺ σ via the twisted spectral characters.
Some Associative Algebras Related to U( ) and Twisted Generalized Weyl Algebras
2003
We prove that both Mickelsson step algebras and Orthogonal Gelfand-Zetlin algebras are twisted generalized Weyl algebras. Using an analogue of the Shapovalov form we construct all weight simple graded modules and some classes of simple weight modules over a twisted generalized Weyl algebra, improving the results from [6], where a particular class of algebras was considered and only special modules were classified.
Finite-dimensional representation theory of loop algebras: a survey
Contemporary Mathematics, 2010
We survey some important results concerning the finite-dimensional representations of the loop algebras g ⊗ C t ±1 of a simple complex Lie algebra g, and their twisted loop subalgebras. In particular, we review the parametrization and description of the Weyl modules and of the irreducible finite-dimensional representations of such algebras, describe a block decomposition of the (non-semisimple) category of their finite-dimensional representations, and conclude with recent developments in the representation theory of multiloop algebras.
Weyl Modules for Classical and Quantum Affine algebras
2000
We define a family of universal finite-dimensional highest weight modules for affine Lie algebras, we call these Weyl modules. We conjecture that these are the classical limits of the irreducible finite--dimensional representations of the quantum affine algebras. We prove this conjecture in the case of affine sl_2. We establish a criterion for these modules to be irreducible and prove a factorization theorem for them in the general case.
Selecta Mathematica, 2011
We propose a notion of algebra of twisted chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of "smallest" such modules are irreducibleĝ-modules and all irreducible gintegrableĝ-modules at the critical level arise in this way. X ⊗H X , where H X is the algebra of differential polynomials on H 1 (X, Ω 1 X → Ω 2,cl X ). Apart from serving as a prototype, algebras of twisted differential operators are directly linked to algebras of twisted chiral differential operators via the notion of the Zhu algebra [Zhu], and this is another topic of the present paper. Zhu attached to each graded vertex algebra V an associative algebra, Zhu(V ). We show that the sheaf associated to the presheaf X ⊃ U → Zhu(D ch,tw X (U )) is precisely D tw X . Zhu(V ) controls representation theory of V , the subject to which we now turn.
Representations of Affine and Toroidal Lie Algebras
2010
We discuss the category calI\cal IcalI of level zero integrable representations of loop algebras and their generalizations. The category is not semisimple and so one is interested in its homological properties. We begin by looking at some approaches which are used in the study of other well--known non--semisimple categories in the representation theory of Lie algebras. This is done with a view to seeing if and how far these approaches can be made to work for calI\cal IcalI. In the later sections we focus first on understanding the irreducible level zero modules and later on certain universal modules, the local and global Weyl modules which in many ways play a role similar to the Verma modules in the BGG--category calO\cal OcalO. In the last section, we discuss the connections with the representation theory of finite--dimensional associative algebras and on some recent work with J. Greenstein.