Three contributions in representation theory

2010

We study the problem of determining if the braid group representations obtained from quantum groups of types $E, F$ and $G$ at roots of unity have infinite image or not. In particular we show that when the fusion categories associated with these quantum groups are not weakly integral, the braid group images are infinite. This provides further evidence for a

Communications in Mathematical Physics, 1993

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups U q (g). They have the same FRT generators l ± but a matrix braided-coproduct ∆L = L⊗L where L = l + Sl − , and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices BM q (2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum double D(U q (sl 2)) (also known as the 'quantum Lorentz group') is the semidirect product as an algebra of two copies of U q (sl 2), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.

Selecta Mathematica, 2015

The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field Fq and any sequence i of simple objects in C the element X V,i of the corresponding algebra P C,i of q-polynomials. We prove that if C was hereditary, then the assignments V → X V,i define algebra homomorphisms from the (dual) Hall-Ringel algebra of C to the P C,i , which generalize the well-known Feigin homomorphisms from the upper half of a quantum group to q-polynomial algebras. If C is the representation category of an acyclic valued quiver (Q, d) and i = (i 0 , i 0), where i 0 is a repetition-free source-adapted sequence, then we prove that the i-character X V,i equals the quantum cluster character X V introduced earlier by the second author in [30] and [31]. Using this identification, we deduce a quantum cluster structure on the quantum unipotent cell corresponding to the square of a Coxeter element. As a corollary, we prove a conjecture from the joint paper [5] of the first author with A. Zelevinsky for such quantum unipotent cells. As a byproduct, we construct the quantum twist and prove that it preserves the triangular basis introduced by A. Zelevinsky and the first author in [6]. 2 A. BERENSTEIN and D. RUPEL 7.2. Quantum twist and Proof of Theorem 2.10 37 7.3. Proof of Theorem 2.11 40 8. Example 40 9. Appendix: Twisted Bialgebras in Braided Monoidal Categories 42 References 45

Letters in Mathematical Physics, 1991

A new quantum group is derived from a 'nonstandard' braid group representation by employing the Faddeev-Reshetlkhin-Takhtajan constructive method. The classical limit is not a Lie superalgebra, despite relations like .,c 2 = )'~ = 0. We classify all finite-dimensional irreducible representations of the new Hopf algebra and find only one-and two-dimensional ones.

We analyze the elements characterizing the theory of induced representations of Lie groups, in order to generalize it to quantum groups. We emphasize the geometric and algebraic aspects of the theory, because they are more suitable for generalization in the framework of Hopf algebras. As an example, we present the induced representations of a quantum deformation of the extended Galilei algebra in (1 + 1) dimensions.

Advances in Mathematics, 2009

We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasi-Yetter-Drinfeld (QYD) modules over Hopf algebras which turn out to be a generalisation of the ordinary Yetter-Drinfeld modules. To each braiding (a solution to the braid equation) we associate a QYD-module and the corresponding braided Heisenberg double -this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by Nichols-Woronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds into the braided Heisenberg double attached to the corresponding complex reflection group.

Pacific Journal of Mathematics, 1995

This article continues the study of concrete algebra-like structures in our polyadic approach , when the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions. In this way, the associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, the dimension of the algebra can be not arbitrary, but " quantized " ; the polyadic convolution product and bialgebra can be defined, when algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to the quantum group theory, we introduce the polyadic version of the braidings, almost co-commutativity, quasitriangularity and the equations for R-matrix (that can be treated as polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.

Transactions of the American Mathematical Society, 2008

Cornell University - arXiv, 2017

Let g be a complex semisimple Lie algebra, τ a point in the upper half-plane, and ∈ C a deformation parameter such that the image of in the elliptic curve C/(Z + τ Z) is of infinite order. In this paper, we give an intrinsic definition of the category of finite-dimensional representations of the elliptic quantum group E ,τ (g) associated to g. The definition is given in terms of Drinfeld half-currents, and extends that given by Enriquez-Felder for g = sl 2 [14]. When g = sln, it reproduces Felder's RLL definition [24] via the Gauß decomposition obtained in [14] for n = 2 and [31] for n ≥ 3. We classify the irreducible representations of E ,τ (g) in terms of elliptic Drinfeld polynomials, in close analogy to the case of the Yangian Y (g) and quantum loop algebra Uq(Lg) of g. A crucial ingredient in the classification is a functor Θ from finitedimensional representations of Uq(Lg) to those of E ,τ (g), which is an elliptic analogue of the monodromy functor constructed in [33], and circumvents the fact that E ,τ (g) does not admit Verma modules. Our classification is new even for g = sl 2. It holds more generally when g is a symmetrisable Kac-Moody algebra, provided finite-dimensionality is replaced by an integrability and category O conditions.