Littelmann paths for the basic representation of an affine Lie algebra

Magyar, Péter

doi:10.1016/j.jalgebra.2006.02.009

Cited by 11 publications

(16 citation statements)

References 27 publications

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“…For G 2 , such a construction has been given by Yamane [21]. For the Lie algebras of type E 6 and E 7 a construction (only for the case Λ = Λ 0 ) was given by Peter Magyar [16] using the path model.…”

Section: Introductionmentioning

confidence: 99%

Tensor Product Structure of Affine Demazure Modules and Limit Constructions

Fourier¹,

Littelmann²

2006

Nagoya Mathematical Journal

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Abstract. Let g be a simple complex Lie algebra, we denote by b g the affine Kac-Moody algebra associated to the extended Dynkin diagram of g. Let Λ0 be the fundamental weight of b g 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 ∨ = P j λ ∨ j as a sum of dominant integral g-coweights, the Demazure module is (as g-module) isomorphic toFor 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 U q (b g)-modules (Kirillov-Reshetikhin-modules) suggests furthermore that all quantized Demazure modules V −λ ∨ ,q (mΛ0) can be naturally endowed with the structure of a U q (b g)-module. We prove, in the classical case (and for a lot of non-classical cases), a conjecture by Kashiwara [10], that the "smallest" Demazure modules are, when viewed as g-modules, isomorphic to some KR-modules. For an integral dominant b g-weight Λ let V (Λ) be the corresponding irreducible b g-representation. Using the tensor product decomposition for Demazure modules, we give a description of the g-module structure of V (Λ) as a semi-infinite tensor product of finite dimensional g-modules. The case of twisted affine Kac-Moody algebras can be treated in the same way, some details are worked out in the last section.

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Section: Introductionmentioning

confidence: 99%

Tensor Product Structure of Affine Demazure Modules and Limit Constructions

Fourier¹,

Littelmann²

2006

Nagoya Mathematical Journal

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“…For exceptional types, we list up all monomials. Most of them have been calculated already in the literature ( [14], [38], [24], [12], [36], [27], [37], [3]), but we have a few new examples in exceptional types. And our method works for arbitrary fundamental representations in principle, though we certainly need to use a computer with huge memory for the triple node of E (1) 8 .…”

Section: Introductionmentioning

confidence: 99%

Level 0 Monomial Crystals

Hernandez¹,

Nakajima²

2006

Nagoya Mathematical Journal

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Abstract. We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of E (2) 6 , E (1) 7 , E (1) 8 . Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of tableaux. For exceptional types, we list up all monomials.

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“…These modules for the current algebra are obtained by taking the current algebra module generated by the extremal vectors in modules of positive level of affine Lie algebras. The dimension (and, actually, character) of these modules was computed in [18,19]. We use these results to prove that the Weyl modules are isomorphic to the Demazure modules in the level one representations of the affine Lie algebra of sl r+1 .…”

Section: Introductionmentioning

confidence: 99%

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

Chari

Loktev

2006

Advances in Mathematics

104

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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.

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Littelmann paths for the basic representation of an affine Lie algebra

Cited by 11 publications

References 27 publications

Tensor Product Structure of Affine Demazure Modules and Limit Constructions

Tensor Product Structure of Affine Demazure Modules and Limit Constructions

Level 0 Monomial Crystals

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

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