Resumen de Weyl, Demazure and fusion modules for the current algebra of

Vyjayanthi Chari, Sergei Loktev

• 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 . 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 , 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.