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 highestweight 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 non-twisted hyper loop algebras, respectively, via restriction of the action. ✩ Partially supported by the FAPESP grant 2007/07456-9 (A. B.) and the CNPq grant 306678/2008-0 (A. M.) Email addresses: angelo@ime.unicamp.br (A. B.), aamoura@ime.unicamp.br (A. M.)
We establish the existence of Demazure flags for graded local Weyl modules for hyper current algebras in positive characteristic. If the underlying simple Lie algebra is simply laced, the flag has length one, i.e., the graded local Weyl modules are isomorphic to Demazure modules. This extends to the positive characteristic setting results of Chari-Loktev, Fourier-Littelmann, and Naoi for current algebras in characteristic zero. Using this result, we prove that the character of local Weyl modules for hyper loop algebras depend only on the highest weight, but not on the (algebraically closed) ground field, and deduce a tensor product factorization for them.
We study the category of Z ℓ -graded modules with finite-dimensional graded pieces for certain Z ℓ + -graded Lie algebras. We also consider certain Serre subcategories with finitely many isomorphism classes of simple objects. We construct projective resolutions for the simple modules in these categories and compute the Ext groups between simple modules. We show that the projective covers of the simple modules in these Serre subcategories can be regarded as multigraded generalizations of Kirillov-Reshetikhin modules and give a recursive formula for computing their graded characters.
We investigate the categories of finite-dimensional representations of multicurrent and multiloop hyperalgebras in positive characteristic, i.e., the hyperalgebras associated to the multicurrent algebras g⊗C[t 1 , . . . , t n ] and to the multiloop algebras g⊗C[t ±1 1 , . . . , t ±1 n ], where g is any finite-dimensional complex simple Lie algebra. The main results are the construction of the universal finite-dimensional highest-weight modules and a classification of irreducible modules in each category. In the characteristic zero setting we also provide a relationship between them.A.B. is partially supported by CNPq grant 462315/2014-2 and FAPESP grants 2015/22040-0 and 2014/09310-5. 1 REPRESENTATIONS OF HYPER MULTICURRENT AND MULTILOOP ALGEBRAS 2The goal of this paper is to establish basic results about the finite-dimensional representations of multicurrent and multiloop hyperalgebras, which have the form g ⊗ C[t 1 , . . . , t n ] and g ⊗ C[t ±1 1 , . . . , t ±1 n ] respectively, extending some of the known results in the case of current and loop algebras (g ⊗ C[t] and g ⊗ C[t ±1 ]). The approach for this is similar to [17,6], with the remark that the known characteristic zero methods as those compiled in [9] are not available for the hyperalgebra setting. We denote the multicurrent algebra in n variables by g[n] and the multiloop algebra by g n . Our main results are the construction of the universal finite-dimensional highest-weight modules in the categories of finite-dimensional
Abstract. We begin the study of a tilting theory in certain truncated categories of modules G(Γ) for the current Lie algebra associated to a finite-dimensional complex simple Lie algebra, where Γ = P + × J, J is an interval in Z, and P + is the set of dominant integral weights of the simple Lie algebra. We use this to put a tilting theory on the category G(Γ ) where Γ = P × J, where P ⊆ P + is saturated. Under certain natural conditions on Γ , we note that G(Γ ) admits full tilting modules.
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