Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions (original) (raw)

Weyl, Demazure and fusion modules for the current algebra of

Advances in Mathematics, 2006

We construct a Poincaré-Birkhoff-Witt type basis for the Weyl modules [V. Chari, A. Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001) 191-223, math.QA/0004174] of the current algebra of sl r+1 . As a corollary we prove the conjecture made in [V. Chari, A. Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001) 191-223, math.QA/0004174; V. Chari, A. Pressley, Integrable and Weyl modules for quantum affine sl 2 , in: Quantum Groups and Lie Theory, Durham, 1999, in: London Math. Soc. Lecture Note Ser., vol. 290, Cambridge Univ. Press, Cambridge, 2001, pp. 48-62, math.QA/0007123] on the dimension of the Weyl modules in this case. Further, we relate the Weyl modules to the fusion modules defined in [B. Feigin, S. Loktev, On generalized Kostka polynomials and the quantum Verlinde rule, in: Differential Topology, Infinite-dimenmath.QA/9812093] of the current algebra and the Demazure modules in level one representations of the corresponding affine algebra. In particular, this allows us to establish substantial cases of the conjectures in [B. Feigin, S. Loktev, On generalized Kostka polynomials and the quantum Verlinde rule, in: Differential Topology, Infinite-dimensional Lie Algebras, and Applications, in: Amer. Math. Soc. Transl. Ser. 2, vol. 194, 1999, pp. 61-79, math.QA/9812093] on the structure and graded character of the fusion modules.

Weyl, Demazure and fusion modules for the current algebra of sl r + 1

Advan Math, 2006

We construct a Poincaré–Birkhoff–Witt type basis for the Weyl modules [V. Chari, A. Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001) 191–223, math.QA/0004174] of the current algebra of slr+1slr+1. As a corollary we prove the conjecture made in [V. Chari, A. Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001) 191–223, math.QA/0004174; V. Chari, A. Pressley, Integrable and Weyl modules for quantum affine sl2sl2, in: Quantum Groups and Lie Theory, Durham, 1999, in: London Math. Soc. Lecture Note Ser., vol. 290, Cambridge Univ. Press, Cambridge, 2001, pp. 48–62, math.QA/0007123] on the dimension of the Weyl modules in this case. Further, we relate the Weyl modules to the fusion modules defined in [B. Feigin, S. Loktev, On generalized Kostka polynomials and the quantum Verlinde rule, in: Differential Topology, Infinite-dimensional Lie Algebras, and Applications, in: Amer. Math. Soc. Transl. Ser. 2, vol. 194, 1999, pp. 61–79, math.QA/9812093] of the current algebra and the Demazure modules in level one representations of the corresponding affine algebra. In particular, this allows us to establish substantial cases of the conjectures in [B. Feigin, S. Loktev, On generalized Kostka polynomials and the quantum Verlinde rule, in: Differential Topology, Infinite-dimensional Lie Algebras, and Applications, in: Amer. Math. Soc. Transl. Ser. 2, vol. 194, 1999, pp. 61–79, math.QA/9812093] on the structure and graded character of the fusion modules.

An application of free Lie algebras to current algebras and their representation theory

2005

We realize the current algebra of a Kac-Moody algebra as a quotient of a semi-direct product of the Kac-Moody Lie algebra and the free Lie algebra of the Kac-Moody algebra. We use this realization to study the representations of the current alg ebra. In particular we see that every ad-invariant ideal in the symmetric algebra of the Kac-Moody algebra gives rise in a canonical way to a representation of the current algebra. These representations include certain well-known families of representations of the current algebra of a simple Lie algebra. Another family of examples, which are the classical limits of the Kirillov-Reshe tikhin modules, are also obtained explicitly by using a construction of Kostant. Finally we study extensi ons in the category of finite dimensional modules of the current algebra of a simple Lie algebra.

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.

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.

Weyl Modules for the Hyperspecial Current Algebra

International Mathematics Research Notices, 2014

ABSTRACT We develop the theory of global and local Weyl modules for the hyperspecial maximal parabolic subalgebra of type A2n(2)A_{2n}^{(2)}A2n(2). We prove that the dimension of a local Weyl module depends only on its highest weight, thus establishing a freeness result for global Weyl modules. Furthermore, we show that the graded local Weyl modules are level one Demazure modules for the corresponding affine Lie algebra. In the last section we derive the same results for the special maximal parabolic subalgebras of the twisted affine Lie algebras not of type A2n(2)A_{2n}^{(2)}A2n(2).

Tensor Product Structure of Affine Demazure Modules and Limit Constructions

Nagoya Mathematical Journal

Let g be a simple complex Lie algebra, we denote by ĝ the affine Kac-Moody algebra associated to the extended Dynkin diagram of g. Let Λ0 be the fundamental weight of ĝ corresponding to the additional node of the extended Dynkin diagram. For a dominant integral g-coweight λ∨, the Demazure submodule V_λ∨ (mΛ0) is a g-module. We provide a description of the g-module structure as a tensor product of “smaller” Demazure modules. More precisely, for any partition of λ∨ = λ∑j as a sum of dominant integral g-coweights, the Demazure module is (as g-module) isomorphic to ⊗j V_ (mΛ0). For the “smallest” case, λ∨ = ω∨ a fundamental coweight, we provide for g of classical type a decomposition of V_ω∨(mΛ0) into irreducible g-modules, so this can be viewed as a natural generalization of the decomposition formulas in [13] and [16]. A comparison with the Uq (g)-characters of certain finite dimensional -modules (Kirillov-Reshetikhin-modules) suggests furthermore that all quantized Demazure modules V_...

Demazure modules, Fusion products and Q--systems

arXiv (Cornell University), 2013

In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the current algebra associated to a simple Lie algebra. These modules are indexed by an |R + |-tuple of partitions ξ = (ξ α), where α varies over a set R + of positive roots of g and we assume that they satisfy a natural compatibility condition. In the case when the ξ α are all rectangular, for instance, we prove that these modules are Demazure modules in various levels. As a consequence we see that the defining relations of Demazure modules can be greatly simplified. We use this simplified presentation to relate our results to the fusion products, defined in [15], of representations of the current algebra. We prove that the Q-system of [22] extends to a canonical short exact sequence of fusion products of representations associated to certain special partitions ξ. Finally, in the last section we deal with the case of sl2 and prove that the modules we define are just fusion products of irreducible representations of the associated current algebra and give monomial bases for these modules.

Weyl modules for the twisted loop algebras

Journal of Algebra, 2008

The notion of a Weyl module, previously defined for the untwisted affine algebras, is extended here to the twisted affine algebras. We describe an identification of the Weyl modules for the twisted affine algebras with suitably chosen Weyl modules for the untwisted affine algebras. This identification allows us to use known results in the untwisted case to compute the dimensions and characters of the Weyl modules for the twisted algebras.