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Dijana Jakelic

Publications and source records attributed to Dijana Jakelic.

6 recordsLinked to original sources
arXiv2010

Tensor products, characters, and blocks of finite-dimensional representations of quantum affine algebras at roots of unity

We establish several results concerning tensor products, q-characters, and the block decomposition of the category of finite-dimensional representations of quantum affine algebras in the root of unity setting. In the generic case, a Weyl module is isomorphic to a tensor product of fundamental representations and this isomorphism was essential for establishing the block decomposition theorem. This is no longer true in the root of unity setting. We overcome the lack of such a tool by utilizing results on specialization of modules. Furthermore, we establish a sufficient condition for a Weyl module to be a tensor product of fundamental representations and prove that this condition is also necessary when the underlying simple Lie algebra is sl(2). We also study the braid group invariance of q-characters of fundamental representations.

math.QA↗
arXiv2008

Finite-Dimensional Representations of Hyper Loop Algebras Over Non-Algebraically Closed Fields

We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change, tensor products of irreducible and Weyl modules, and the block decomposition of the underlying abelian category. Several results are interestingly related to the study of irreducible representations of polynomial algebras and Galois theory.

math.RT↗
arXiv2007

Finite-Dimensional Representations of Hyper Loop Algebras

We study finite-dimensional representations of hyper loop algebras, i.e., 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.

math.RT↗
arXiv2007

On Crystal Bases and Enright's Completions

We investigate the interplay of crystal bases and completions in the sense of Enright on certain nonintegrable representations of quantum groups. We define completions of crystal bases, show that this notion of completion is compatible with Enright's completion of modules, prove that every module in our category has a crystal basis which can be completed and that a completion of the crystal lattice is unique. Furthermore, we give two constructions of the completion of a crystal lattice.

math.QA↗