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Weyl, Demazure and fusion modules for the current algebra of \(\mathfrak{sl}_{r+1}\). (English) Zbl 1161.17318
Summary: We construct a PoincarĂ©-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)
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