# zbMATH — the first resource for mathematics

Weyl, Demazure and fusion modules for the current algebra of $$\mathfrak{sl}_{r+1}$$. (English) Zbl 1161.17318
Summary: We construct a Poincaré-Birkhoff-Witt type basis for the Weyl modules [V. Chari and A. Pressley, Represent. Theory 5, 191–223 (2001; Zbl 0989.17019), math.QA/0004174] of the current algebra of $$\mathfrak{sl}_{r+1}$$. As a corollary we prove the conjecture made in [V. Chari, A. Pressley, loc. cit. and in: Quantum Groups and Lie Theory, Durham, 1999, Lond. Math. Soc. Lect. Note Ser. 290, 48–62 (2001; Zbl 1034.17008); cf. also \urlmath.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 and S. Loktev, Transl., Ser. 2, Am. Math. Soc. 194(44), 61–79 (1999; Zbl 0974.17008); cf. also \urlmath.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, loc. cit.] on the structure and graded character of the fusion modules.

##### MSC:
 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
Full Text:
##### References:
 [1] Chari, V.; de Moura, A., Characters and blocks for finite-dimensional representations of quantum affine algebras, Int. math. res. not., 5, 257-298, (2005) · Zbl 1074.17004 [2] Chari, V.; Pressley, A., A new family of irreducible integrable representations of affine Lie algebras, Math. ann., 277, 3, 543-562, (1987) · Zbl 0608.17009 [3] Chari, V.; Pressley, A., Weyl modules for classical and quantum affine algebras, Represent. theory, 5, 191-223, (2001) · Zbl 0989.17019 [4] Chari, V.; Pressley, A., Integrable and Weyl modules for quantum affine $$\mathfrak{sl}_2$$, (), 48-62 · Zbl 1034.17008 [5] Feigin, B.; Kirillov, A.N.; Loktev, S., Combinatorics and geometry of higher level Weyl modules, preprint · Zbl 1129.22011 [6] Feigin, B.; Loktev, S., On generalized kostka polynomials and the quantum Verlinde rule, (), 61-79 · Zbl 0974.17008 [7] Fourier, G.; Littelmann, P., Tensor product structure of affine Demazure modules and limit constructions · Zbl 1143.22010 [8] Fourier, G.; Littelmann, P., Weyl modules, affine Demazure modules, fusion products and limit constructions · Zbl 1114.22010 [9] Frenkel, E.; Mukhin, E., Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, Comm. math. phys., 216, 1, 23-57, (2001) · Zbl 1051.17013 [10] Frenkel, E.; Reshetikhin, N., The q-characters of representations of quantum affine algebras and deformations of W-algebras, (), 163-205 · Zbl 0973.17015 [11] Gelfand, I.M.; Tsetlin, M.L., Finite-dimensional representations of the group of unimodular matrices, Dokl. akad. nauk USSR, 71, 825-828, (1950) · Zbl 0037.15301 [12] Hatayama, G.; Kirillov, A.N.; Kuniba, A.; Okado, M.; Takagi, T.; Yamada, Y., Character formulae of $$\hat{\mathfrak{sl}_n}$$-modules and inhomogeneous paths, Nuclear phys. B, 536, 575-616, (1999) · Zbl 0952.17013 [13] Hatayama, G.; Kuniba, A.; Okado, M.; Takagi, T.; Tsuboi, Z., Paths, crystals and fermionic formulae, Prog. math. phys., 23, 205-272, (2002) · Zbl 1016.17011 [14] Kac, V., Infinite dimensional Lie algebras, (1985), Cambridge Univ. Press Cambridge [15] Kashiwara, M., On level-zero representation of quantized affine algebras, Duke math. J., 112, 1, 117-195, (2002) · Zbl 1033.17017 [16] Kedem, R., Fusion products, cohomology of $$\mathit{GL}(N)$$ flag manifolds and kostka polynomials, Int. math. res. not., 25, 1273-1298, (2004) · Zbl 1074.17003 [17] Macdonald, I., Symmetric functions and Hall polynomials, Oxford math. monogr., (1979), The Clarendon Press, Oxford Univ. Press New York · Zbl 0487.20007 [18] Kuniba, A.; Misra, K.C.; Okado, M.; Takagi, T.; Uchiyama, J., Paths, Demazure crystals and symmetric functions, J. math. phys., 41, 9, 6477-6486, (2000) · Zbl 0971.17006 [19] Magyar, P., Littelmann’s paths for the basic representations of an affine Lie algebra [20] Nakajima, H., Quiver varieties and finite-dimensional representations of quantum affine algebras, J. amer. math. soc., 14, 1, 145-238, (2001) · Zbl 0981.17016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.