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Abstract
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We study finite-dimensional
representations of hyper loop algebras, that is, the hyperalgebras over an
algebraically closed field of positive characteristic associated to the loop algebra over
a complex finite-dimensional simple Lie algebra. The main results are the
classification of the irreducible modules, a version of Steinberg’s tensor product
theorem, and the construction of positive characteristic analogues of the Weyl
modules as defined by Chari and Pressley in the characteristic zero setting.
Furthermore, we start the study of reduction modulo p and prove that every
irreducible module of a hyper loop algebra can be constructed as a quotient of a
module obtained by a certain reduction modulo p process applied to a suitable
characteristic zero module. We conjecture that the Weyl modules are also obtained
by reduction modulo p. The conjecture implies a tensor product decomposition for
the Weyl modules which we use to describe the blocks of the underlying abelian
category.
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Keywords
loop algebras, finite-dimensional
representations, hyperalgebras
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Mathematical Subject Classification
Primary: 17B10, 17B65
Secondary: 20G42
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Milestones
Received: 11 December 2006
Revised: 11 September 2007
Accepted: 11 September 2007
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