The classification of finite-dimensional pointed Hopf algebras with group S_3 was finished in "The Nichols algebra of a semisimple Yetter-Drinfeld module", arXiv:0803.2430v1 [math.QA], by Andruskiewitsch, Heckenberger and Schneider: there are exactly two of them, the bosonization of a Nichols algebra of dimension 12 and a non-trivial lifting. Here we determine all simple modules over any of these Hopf algebras. We also find the Gabriel quivers, the projective covers of the simple modules, and prove that they are not of finite representation type. To this end, we first investigate the modules over some complex pointed Hopf algebras defined in the papers "Examples of liftings of Nichols algebras over racks", by Andruskiewitsch and Gra\~na and "Finite dimensional pointed Hopf algebras over S_4", arXiv:0904.2558v2 [math.QA], by G. Garcia and the author, whose restriction to the group of group-likes is a direct sum of 1-dimensional modules.
Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych
et al.
This work is a detailed version of arXiv:0704.0195 [math.QA]. We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also give a comprehensive and self-contained exposition of the known results on braided fusion categories without assuming them pre-modular or non-degenerate. The guiding heuristic principle of our work is an analogy between braided fusion categories and Casimir Lie algebras.
We discussed twisted quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S.Zakrzewski classification can be presented as a sum of subordinated r-matrices of Abelian and Jordanian types. Corresponding twists describing quantum deformations are obtained in explicit form. This work is an extended version of the paper \url{arXiv:0704.0081v1 [math.QA]}.
In our earlier work math.QA/9808015 some results on integral representations of functions in quantum disc were announced. It was then shown in math.QA/9808037 that the validity of those results is related to the invariance of kernels of some integral operators. We introduce here a method which allows us to prove the invariance of the above kernels.
Recall that a finite group is called perfect if it does not have non-trivial 1-dimensional representations (over the field of complex numbers C). By analogy, let us say that a finite dimensional Hopf algebra H over C is perfect if any 1-dimensional H-module is trivial. Let us say that H is biperfect if both H and H^* are perfect. Note that, H is biperfect if and only if its quantum double D(H) is biperfect. It is not easy to construct a biperfect Hopf algebra of dimension >1. The goal of this note is to describe the simplest such example we know. The biperfect Hopf algebra H we construct is based on the Mathiew group of degree 24, and it is semisimple. Therefore, it yields a negative answer to Question 7.5 from a previous paper of the first two authors (math.QA/9905168). Namely, it shows that Corollary 7.4 from this paper stating that a triangular semisimple Hopf algebra over C has a non-trivial group-like element, fails in the quasitriangular case. The counterexample is the quantum double D(H).
Abstract The q -characters were introduced by Frenkel and Reshetikhin [The q -characters of representations of quantum affine algebras and deformations of W -algebras, in: Recent Developments in Quantum Affine Algebras and Related Topics, in: Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 163–205] to study finite dimensional representations of the untwisted quantum affine algebra U q ( g ) for q generic. The e -characters at roots of unity were constructed by Frenkel and Mukhin [The q -characters at roots of unity, Adv. Math. 171 (1) (2002) 139–167] to study finite dimensional representations of various specializations of U q ( g ) at q s =1. In the finite simply laced case Nakajima [ t -analogue of the q -characters of finite dimensional representations of quantum affine algebras, in: Physics and Combinatorics, Proc. Nagoya 2000 Internat. Workshop, World Scientific, Singapore, 2001, pp. 181–212; Quiver varieties and t -analogs of q -characters of quantum affine algebras, Ann. of Math., in press; preprint arXiv: math.QA/0105173 ] defined deformations of q -characters called q , t -characters for q generic and also at roots of unity. The definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties which holds only in the simply laced case. In [Algebraic approach to q , t -characters, Adv. Math., in press, preprint arXiv: math.QA/0212257 ] we proposed an algebraic general (non-necessarily simply laced) new approach to q , t -characters for q generic. In this paper we propose two developments of this approach: we treat the root of unity case and the case of a larger class of generalized Cartan matrices (including finite and affine cases except A 1 (1) , A 2 (2) ). In particular, we generalize the construction of analogs of Kazhdan–Lusztig polynomials at roots of unity of [H. Nakajima, Quiver varieties and t -analogs of q -characters of quantum affine algebras, Ann. of Math., in press, preprint arXiv: math.QA/0105173 ] to those cases. We also study properties of various objects used in this article: deformed screening operators at roots of unity, t -deformed polynomial algebras, bicharacters arising from symmetrizable Cartan matrices, deformation of the Frenkel–Mukhin's algorithm.
This work contains a proof of theorem 7.3 from math.QA/9808015. This theorem demonstrates the Berezin method to be applicable for producing a well known one-parameter deformation of the quantum disc.
In math.QA/0311303 B. Feigin, G. Felder, and B. Shoikhet proposed an explicit formula for the trace density map from the quantum algebra of functions on an arbitrary symplectic manifold M to the top degree cohomology of M. They also evaluated this map on the trivial element of K-theory of the algebra of quantum functions. In our paper we evaluate the map on an arbitrary element of K-theory, and show that the result is expressed in terms of the A-genus of M, the Deligne-Fedosov class of the quantum algebra, and the Chern character of the principal symbol of the element. For a smooth (real) symplectic manifold (without a boundary), this result implies the Fedosov-Nest-Tsygan algebraic index theorem.
We show how the machine invented by S. Merkulov [S.A. Merkulov, Operads, deformation theory and F-manifolds, in: Frobenius Manifolds, Aspects Math., vol. E36, Vieweg, Wiesbaden, 2004, pp. 213–251; S.A. Merkulov, PROP profile of deformation quantization, Preprint, math.QA/0412257, December 2004; S.A. Merkulov, PROP profile of Poisson geometry, Comm. Math. Phys. 262 (1) (February 2006) 117–135] can be used to study and classify natural operators in differential geometry. We also give an interpretation of graph complexes arising in this context in terms of representation theory. As application, we prove several results on classification of natural operators acting on vector fields and connections.
We use the author's combinatorial theory of full heaps (defined in math.QA/0605768) to categorify the action of a large class of Weyl groups on their root systems, and thus to give an elementary and uniform construction of a family of faithful permutation representations of Weyl groups. Examples include the standard representations of affine Weyl groups as permutations of ${\Bbb Z}$ and geometrical examples such as the realization of the Weyl group of type $E_6$ as permutations of 27 lines on a cubic surface; in the latter case, we also show how to recover the incidence relations between the lines from the structure of the heap. Another class of examples involves the action of certain Weyl groups on sets of pairs $(t, f)$, where $t \in {\Bbb Z}$ and $f$ is a function from a suitably chosen set to the two-element set $\{+, -\}$. Each of the permutation representations corresponds to a module for a Kac--Moody algebra, and gives an explicit basis for it.
It has been shown recently, in a joint work with Michel Dubois-Violette and Marc Wambst (see math.QA/0203035), that Koszul property of $N$-homogeneous algebras (as defined in the original paper) becomes natural in a $N$-complex setting. A basic question is to define the differential of the bimodule Koszul complex of an $N$-homogeneous algebra, e.g., for computing its Hochschild homology. The differential defined here uses $N$-complexes. That puts right the wrong differential presented in the original paper in a 2-complex setting. Actually, as we shall see, it is impossible to avoid $N$-complexes in defining the differential, whereas the bimodule Koszul complex is a 2-complex.