Finite-dimensional Representations Of Twisted Hyper-loop Algebras

We investigate the category of finite-dimensional representations of twisted hyper-loop algebras, i.e., the hyperalgebras associated to twisted loop algebras over finite-dimensional simple Lie algebras. The main results are the classification of the irreducible modules, the definition of the universal highest-weight modules, called the Weyl modules, and, under a certain mild restriction on the characteristic of the ground field, a proof that the simple modules and the Weyl modules for the twisted hyper-loop algebras are isomorphic to appropriate simple and Weyl modules for the nontwisted hyper-loop algebras, respectively, via restriction of the action. © 2014 Copyright Taylor & Francis Group, LLC.42731473182Beck, J., Nakajima, H., Crystal bases and two-sided cells of quantum affine algebras (2004) Duke Math. J., 123, pp. 335-402Bianchi, A., (2012) Representações de hiperálgebras de laços e álgebras de multicorrentes, , Ph.D. Thesis, Unicamp, Campinas, BrazilBianchi, A., Macedo, T., Moura, A., On Demazure and local Weyl modules for affine hyperalgebras, , Preprint:arXiv:1307.4305Chari, V., Fourier, G., Senesi, P., Weyl Modules for the twisted loop algebras (2008) J. Algebra, 319 (12), pp. 5016-5038Chari, V., Loktev, S., Weyl, Demazure and fusion modules for the current algebra of s{fraktur}l{fraktur}r+1 (2006) Adv. in Math., 207 (2), pp. 928-960Chari, V., Pressley, A., Weyl modules for classical and quantum affine algebras (2001) Represent. Theory, 5, pp. 191-223Efrat, I., (2006) Valuation, Orderings, and Milnor K-Theory, , Mathematical Surveys and Monographs AMS, 124Fourier, G., Khandai, T., Kus, D., Savage, A., Local Weyl modules for equivariant map algebras with free abelian group actions (2012) J. Algebra, 350, pp. 386-404Fourier, G., Kus, D., Demazure modules and Weyl modules: The twisted current case (2013) To appear in Trans. Amer. Math. Soc., 365 (11), pp. 6037-6064Fourier, G., Littelmann, P., Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions (2007) Adv. in Math., 211 (2), pp. 566-593Fourier, G., Manning, N., Senesi, P., Global Weyl modules for the twisted loop algebra (2013) Abh. Math. Semin. Univ. Hambg., 83 (1), pp. 53-82Garland, H., The arithmetic theory of loop algebras (1978) J. Algebra, 53, pp. 480-551Humphreys, J.E., (1970) Introduction to Lie algebras and representation theory, , Springer-Verlag, GTM, 9Jakelic, D., Moura, A., Finite-dimensional representations of hyper-loop algebras (2007) Pacific J. Math., 233 (2), pp. 371-402Jakelic, D., Moura, A., Finite-dimensional representations of hyper-loop algebras over non-algebraically closed fields (2010) Algebras and Representation Theory, 13 (3), pp. 271-301Jakelic, D., Moura, A., On multiplicity problems for finite-dimensional representations of hyper-loop algebras (2009) Contemp. Math., 483, pp. 147-159Jantzen, J., (1987) Representations of Algebraic Groups, , Boston, Academic PressKac, V., (1990) Infinite Dimensional Lie Algebras, , New York, Cambridge University PressKashiwara, M., Crystal bases of modified quantized enveloping algebras (1994) Duke Math. J., 73, pp. 383-413Kashiwara, M., On level zero representations of quantized affine algebras (2002) Duke Math. J., 112, pp. 117-195Kostant, B., (1966) Groups over ℤ, , Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math. IX, Providence, AMSMitzman, D., Integral bases for affine lie algebras and their universal enveloping algebras (1983) Contemp. Math., p. 40Nakajima, H., Quiver varieties and finite-dimensional representations of quantum affine algebras (2001) J. Amer. Math. Soc., 14, pp. 145-238Nakajima, H., Extremal weight modules of quantum affine algebras (2004) Adv. Stud. Pure Math., 40, pp. 343-369Naoi, K., Weyl modules, Demazure modules and finite crystals for non-simply laced type (2012) Adv. in Math., 229 (2), pp. 875-934Neher, E., Savage, A., Extensions and block decompositions for finite-dimensional representations of equivariant map algebras, , arXiv:1103.4367Neher, E., Savage, A., Senesi, P., Irreducible finite-dimensional representations of equivariant map algebras (2012) Trans. Amer. Math. Soc., 364 (5), pp. 2619-2646Prevost, S., (1992) Vertex algebras and integral Bases for the enveloping algebras of affine lie algebras, , Mem. Math. Amer. Soc., 466Senesi, P., The block decomposition of finite-dimensional representations of twisted loop algebras (2010) Pacific J. Math., 244 (2), pp. 355-357Serre, J.-P., (1980) Local Fields, , Springer-Verlag GTM 6