Braided bosonization and inhomogeneous quantum groups

We consider quasitriangular Hopf algebras in braided tensor categories introduced by Majid. It is known that a quasitriangular Hopf algebra H in a braided monoidal category C induces a braiding in a full monoidal subcategory of the category of H-modules in C. Within this subcategory, a braided version of the bosonization theorem with respect to the category C will be proved. An example of braided monoidal categories with quasitriangular structure deviating from the ordinary case of symmetric tensor categories of vector spaces is provided by certain braided supersymmetric tensor categories. Braided inhomogeneous quantum groups like the dilaton free q-Poincaré group are explicit applications.Supported in part by the Deutsche Forschungsgemeinschaft (DFG) through a research fellowship.     

Quantum groups at roots of ±1

Apart from being of interest in its own right, the representation theory for quantum groups at roots of unity enters into Lusztig’s programme (see e.g. [Lus94]) for determining the irreducible characters of semi-simple algebraic groups in characteristic p > 0. In [AJS94] this connection plays a key role. There our assumption is that the root of unity has odd order (bigger than the Coxeter number).

Using the recent work of Kashiwara - Tanisaki, [KT95a] and [KT95b] together with the Kazhdan-Lusztig equivalence of categories, [KL93], [KL94] and [Lus94] it is possible to get rid of this oddness assumption as far as the determination of irreducible characters for quantum groups goes, see also [Kan95]. In this note we show how we can do this much more directly staying inside the theory for quantum groups. To be more specific we prove the following result (see below for more details and notations).     

Pointed Hopf Algebras with Triangular Decomposition

In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (Algebr. Represent. Theory 7(3) ? BC) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free abelian and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to \(\mathfrak {sl}_{2}\).     

Integral lattices in TQFT

We find explicit bases for naturally defined lattices over a ring of algebraic integers in the SO(3)-TQFT-modules of surfaces at roots of unity of odd prime order. Some applications relating quantum invariants to classical 3-manifold topology are given.     

The BLM realization for the integral form of Uv(glm|n)

In this paper, we embed the integral form of the quantum supergroup U_v(gl_(m|n)) to the product of a family of integral quantum Schur super algebras. We show that the image of the embedding is a free Z[v, v~(-1)]-module by finding the basis explicitly and calculating the fundamental multiplication formulas of these bases. Unlike the non-super case, the fundamental multiplication formula, which is the key step, is more complicated since we have to deal with the case of multiplying the odd root vectors. As a consequence, via the base change, we realize the quantum supergroup at roots of unity as a subalgebra of the product of quantum Schur superalgebras. Thus, we find a new basis of quantum supergroups at odd roots of unity which comes from quantum Schur superalgebras.     

Classical and Exceptional Root Systems and Quantizations

We present a method for quantizing semisimple Lie algebras. In Huru (Russ. Math. [2007]) we defined quantizations of the braided Lie algebra structure on a finite dimensional graded vector space V by quantizations of braided derivations on the exterior algebra of V ^* . We find quantizations of semisimple Lie algebras in this setting using the grading by their roots and shall go through all root systems, classical and exceptional.      

Quantum analogues of a coherent family of modules at roots of unity: A 3

To a given coherent family of virtual representations of a complex semiesimple Lie algebra we associate elsewhere a coherent family of virtual representations of the corresponding quantum group at roots of unity. We also proposed a conjecture there that under some hyptoheses the members of the family in a certain positive cone are actually modules (and not just a virtual module, i.e., a difference of two modules). The validity of this conjecture for A ₂ and B ₂ has been verified by the author. In this paper we solve the problem for A ₃.     

Lie algebras and recurrence relations I

We study representations of the Heisenberg-Weyl algebra and a variety of Lie algebras, e.g., su(2), related through various aspects of the spectral theory of self-adjoint operators, the theory of orthogonal polynomials, and basic quantum theory. The approach taken here enables extensions from the one-variable case to be made in a natural manner. Extensions to certain infinite-dimensional Lie algebras (continuous tensor products, q-analogs) can be found as well. Particularly, we discuss the relationship between generating functions and representations of Lie algebras, spectral theory for operators that lead to systems of orthogonal polynomials and, importantly, the precise connection between the representation theory of Lie algebras and classical probability distributions is presented via the notions of quantum probability theory. Coincidentally, our theory is closed connected to the study of exponential families with quadratic variance in statistical theory.     

Module categories over pointed Hopf algebras

We develop some techniques for studying exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the small quantum groups u_q(\mathfraksl₂){u_q(\mathfrak{sl}_2)}.     

Correspondences of ribbon categories

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally, we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.     

Improving two theorems of bose on difference families

In [2] R. C. Bose gives a sufficient condition for the existence of a (q, 5, 1) difference family in (GF(q), +)—where q ≡ 1 mod 20 is a prime power — with the property that every base block is a coset of the 5th roots of unity. Similarly he gives a sufficient condition for the existence of a (q, 4, 1) difference family in (GF(q, +)—where q ≡ 1 mod 12 is a prime power — with the property that every base block is the union of a coset of the 3rd roots of unity with zero. In this article we replace the mentioned sufficient conditions with necessary and sufficient ones. As a consequence, we obtain new infinite classes of simple difference families and hence new Steiner 2-designs with block sizes 4 and 5. In particular, we get a (p^4α, 5, 1)-DF for any odd prime p ≡ 2, 3 (mod 5), and a (p^2α, 4, 1)-DF for any odd prime p ≡ 2 (mod 3). © 1995 John Wiley & Sons, Inc.     

On simple radical difference families

For q a prime power and k odd (even), we define a (q,k,1) difference family to be radical if each base block is a coset of the kth roots of unity in the multiplicative group of GF(q) (the union of a coset of the (k ? 1)th roots of unity in the multiplicative group of GF(q) with zero). Such a family will be denoted by RDF. The main result on this subject is a theorem dated 1972 by R.M. Wilson; it is a sufficient condition for the existence of a (q,k, 1)-RDF for any k. We improve this result by replacing Wilson's condition with another sufficient but weaker condition, which is proved to be necessary at least for k ? 7. As a consequence, we get new difference families and hence new Steiner 2-designs. © 1995 John Wiley & Sons, Inc.     

Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups

The aim of this note is to characterize certain probability laws on a class of quantum groups or braided groups that will be called nilpotent. First, we introduce a braided analogue of the Heisenberg-Weyl group, which shall serve as a standard example. We determine functional on the braided line and on this group satisfying an analogue of the Bernstein property (see [3]). i.e. that the sum and difference of independent Gaussian random variables are also independent. We also study continuous convolution semigroups on nilpotent quantum groups and braided groups. We extend to nilpotent quantum groups and braided groups recent results proving the uniqueness of the embedding of an infinitely divisible probability law in a continuous convolution semigroup for simply connected nilpotent Lie groups.     

Categorical Morita Equivalence for Group-Theoretical Categories

A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the dual of a pointed semisimple category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs (G, ω), where G is a finite group and ω ? H ³(G, k ^×). A group-theoretical and cohomological interpretation of this relation is given. A series of concrete examples of pairs of groups that are categorically Morita equivalent but have nonisomorphic Grothendieck rings are given. In particular, the representation categories of the Drinfeld doubles of the groups in each example are equivalent as braided tensor categories and hence these groups define the same modular data.     

Modified 6j-symbols and 3-manifold invariants

We show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in Geer et al. (2009) [14], lead to modified 6j-symbols and to new state sum 3-manifold invariants. We give examples of categories such that the associated standard Turaev–Viro 3-manifold invariants vanish but the secondary invariants may be non-zero. The categories in these examples are pivotal categories which are neither ribbon nor semi-simple and have an infinite number of simple objects.     

The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras

Let H be a Hopf algebra over the field k which is a finite module over a central affine sub-Hopf algebra R. Examples include enveloping algebras of finite dimensional k-Lie algebras in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity. The ramification behaviour of the maximal ideals of Z(H) with respect to the subalgebra R is studied, and the conclusions are then applied to the cases of classical and quantised enveloping algebras. In the case of for semisimple a conjecture of Humphreys [28] on the block structure of is confirmed. In the case of for semisimple and an odd root of unity we obtain a quantum analogue of a result of Mirković and Rumynin, [35], and we fully describe the factor algebras lying over the regular sheet, [9]. The blocks of are determined, and a necessary condition (which may also be sufficient) for a baby Verma -module to be simple is obtained. Received: 24 June 1999; in final form: 30 March 2000 / Published online: 17 May 2001     

Vertex representation of quantum N-toroidal algebra for type F4

Abstract

Recently Gao-Jing-Xia-Zhang defined the structures of quantum N-toroidal algebras uniformally, which are a kind of natural generalizations of the classical quantum toroidal algebras, just like the relation between 2-toroidal Lie algebras and N-toroidal Lie algebras. Based on this work, we construct a level-one vertex representation of quantum N-toroidal algebra for type F₄. In particular, we can also obtain a level-one vertex representation of quantum toroidal algebra for type F₄ as our special cases.     

Homological sections of Lie algebroids

We examine Lie (super)algebroids equipped with a homological section, i.e., an odd section that ‘self-commutes’, we refer to such Lie algebroids as inner Q-algebroids: these provide natural examples of suitably “superised” Q-algebroids in the sense of Mehta. Such Lie algebroids are a natural generalisation of Q-manifolds and Lie superalgebras equipped with a homological element. Amongst other results, we show that, via the derived bracket formalism, the space of sections of an inner Q-algebroid comes equipped with an odd Loday–Leibniz bracket.     

The centralizer algebras of lie color algebras

In his 1977 paper, V.G. Kac classified the finite-dimensional simple complex Lie superalgebras. After Kac’s paper, M. Scheunert initiated the study of a generalization of Lie superalgebras - the Lie color algebras. We construct some new families of simple Lie color algebras. Following the work of A. Berele and A. Regev and A.N. Sergeev, who studied the general linear and sq(n)-series superalgebra cases, and the work of G. Benkart, C. Lee Shader, and A. Ram, who studied the orthosymplectic cases, we examine the centralizer algebras of some classical Lie superalgebras and their Lie color algebra counterparts acting on tensor space and derive Schur-Weyl duality results for these algebras and their centralizers.