Representations of quantum affine algebras (original) (raw)
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Equivalence of certain categories of modules for quantized affine lie algebras
Journal of The Australian Mathematical Society, 2000
We show that a quantum Verma-type module for a quantum group associated to an affine Kac-Moody algebra is characterized by its subspace of finite-dimensional weight spaces. In order to do this we prove an explicit equivalence of categories between a certain category containing the quantum Verma modules and a category of modules for a subalgebra of the quantum group for which the finite part of the Verma module is itself a module.
Algebraic structures in group-theoretical fusion categories
2020
It was shown by Ostrik (2003) and Natale (2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We generalize this result by constructing explicit Morita equivalence class representatives of indecomposable, separable algebras in group-theoretical fusion categories. This is achieved by providing the `free functor' Phi\PhiPhi from a pointed fusion category to a group-theoretical fusion category with a monoidal structure. Our algebras of interest are then constructed as the image of twisted group algebras under Phi\PhiPhi. We also show that twisted group algebras admit the structure of Frobenius algebras in a pointed fusion category, and we establish a Frobenius monoidal structure on Phi\PhiPhi as well. As a consequence, our algebras are Frobenius algebras in a group-theoretical fusion category, and like twisted group algebras in the pointed case, they a...
Annals of Mathematics, 2005
Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not necessarily hermitian) modular category is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category. At the end of the paper we generalize some of these results to positive characteristic.
Representations of Quantum Affinizations and Fusion Product
Transformation Groups, 2005
In this paper we study general quantum affinizations Uq (ĝ) of symmetrizable quantum Kac-Moody algebras and we develop their representation theory. We prove a triangular decomposition and we give a classication of (type 1) highest weight simple integrable representations analog to Drinfel'd-Chari-Pressley one. A generalization of the q-characters morphism, introduced by Frenkel-Reshetikhin for quantum affine algebras, appears to be a powerful tool for this investigation. For a large class of quantum affinizations (including quantum affine algebras and quantum toroidal algebras), the combinatorics of qcharacters give a ring structure * on the Grothendieck group Rep(Uq (ĝ)) of the integrable representations that we classified. We propose a new construction of tensor products in a larger category by using the Drinfel'd new coproduct (it can not directly be used for Rep(Uq (ĝ)) because it involves infinite sums). In particular we prove that * is a fusion product (a product of representations is a representation).
Quantization of Lie Bialgebras, Part VI: Quantization of Generalized Kac–Moody Algebras
Transformation Groups - TRANSFORM GROUPS, 2008
This paper is a continuation of the series of papers “Quantization of Lie bialgebras (QLB) I-V”. We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular structure. This implies that when the quantization parameter is formal, then the category O for the quantized Kac-Moody algebra is equivalent, as a braided tensor category, to the category O over the corresponding classical Kac-Moody algebra, with the tensor category structure defined by a Drinfeld associator. This equivalence is a generalization of the functor constructed previously by G. Lusztig and the second author. In particular, we answer positively a question of Drinfeld whether the characters of irreducible highest weight modules for quantized Kac-Moody algebras are the same as in the classical case. Moreover, our results are valid...
The Fukaya Type Categories For Associative Algebras
this paper, however, we will show that, if one takes for End A the differential graded algebra C (A; A) of Hochschild cochains, the maps (1), in a sense, still exist. More precisely, they exist if one passes from the category of algebras to the category of complexes by means of some well known homological functors. Let A be an associative unital algebra over a commutative unital ground ring k. Consider A as a bimodule over itself; by E A we denote the differential graded algebra (C (A; A); ffi; ) which is the standard complex for computing Ext A\Omega A ffi (A; A) = H (A; A) (the Hochschild cohomology) equipped with the Yoneda product (Sect. 2). We construct the map of complexes ffl : C (A)\Omega C (E A ) ! C (A) (2) ffl : C (E A )\Omega C (E A ) ! C (E A ) (3) Here C stands for the Hochschild complex computing HH (A) or for the periodic cyclic complex. For the case of bar complex an analogous operation was discovered by Getzler--Jones and by Gerstenhaber--Voronov [11, 9]. Therefore...