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 superalgebra representations on cohomology groups of non-commutative bundles

Quantum homogeneous supervector bundles arising from the quantum general linear supergoup are studied. The space of holomorphic sections is promoted to a left exact covariant functor from a category of modules over a quantum parabolic sub-supergroup to the category of locally finite modules of the quantum general linear supergroup. The right derived functors of this functor provides a form of Dolbeault cohomology for quantum homogeneous supervector bundles. We explicitly compute the cohomology groups, which are given in terms of well understood modules over the quantized universal enveloping algebra of the general linear superalgebra.     

Drinfeld–Sokolov reduction for quantum groups and deformations of W-algebras

We define deformations of W-algebras associated to comple semisimple Lie algebras by means of quantum Drinfeld-Sokolov reduction procedure for affine quantum groups. We also introduce Wakimoto modules for arbitrary affine quantum groups and construct free field resolutions and screening operators for the deformed W-algebras. We compare our results with earlier definitions of q-W-algebras and of the deformed screening operators due to Awata, Kubo, Odake, Shiraishi [60],[6], [7], Feigin, E. Frenkel [22] and E. Frenkel, Reshetikhin [34]. The screening operator and the free field resolution for the deformed W-algebra associated to the simple Lie algebra sl₂ coincide with those for the deformed Virasoro algebra introduced in [60]. The author is supported by the Swiss National Science Foundation.     

Quantum groups and ribbon G-categories

For a group G, the notion of a ribbon G-category was introduced in Turaev (Homotopy field theory in dimension 3 and crossed group-categories, preprint, math. GT/0005291) with a view towards constructing 3-dimensional homotopy quantum field theories (HQFTs) with target K(G,1). We discuss here how to derive ribbon G-categories from a simple complex Lie algebra where G is the center of . Our construction is based on a study of representations of the quantum group at a root of unity ε. Under certain assumptions on ε, the resulting G-categories give rise to numerical invariants of pairs (a closed oriented 3-manifold M, an element of H¹(M;G)) and to 3-dimensional HQFTs.     

Realization of PBW-deformations of type An quantum groups via multiple Ore extensions

The notion of multiple Ore extension is introduced as a natural generalization of Ore extensions and double Ore extensions. For a PBW-deformation ${\mathfrak{B}}_q(\mathfrak{sl}(n+1,\mathbb{C}))$ of type ${\mathbb{A}}_n$ quantum group, we explicitly obtain the commutation relations of its root vectors, then show that it can be realized via a series of multiple Ore extensions, which we call a ladder Ore extension of type $(1,2,?,n)$. Moreover, we analyze the quantum algebras ${\mathfrak{B}}_q(\mathfrak{g})$ with $\mathfrak{g}$ of type ${\mathbb{D}}_4$, ${\mathbb{B}}_2$ and ${\mathbb{G}}_2$ and give some examples and counterexamples that can be realized by a ladder Ore extension.     

Representations of quantum groups defined over commutative rings II

In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules.     

Classical and quantum Heisenberg groups, their representations and applications

This paper is a survey on classical Heisenberg groups and algebras, q-deformed Heisenberg algebras, q-oscillator algebras, their representations and applications. Describing them, we tried, for the reader's convenience, to explain where the q-deformed case is close to the classical one, and where there are principal differences. Different realizations of classical Heisenberg groups, their geometrical aspects, and their representations are given. Moreover, relations of Heisenberg groups to other linear groups are described. Intertwining operators for different (Schrödinger, Fock, compact) realizations of unitary irreducible representations of Heisenberg groups are given in explicit form. Classification of irreducible representations and representations of the q-oscillator algebra is derived for the cases when q is not a root of unity and when q is a root of unity. The Fock representation of the q-oscillator algebra is studied in detail. In particular, q-coherent states are described. Spectral properties of some operators of the Fock representations of q-oscillator algebras are given. Some of applications of Heisenberg groups and algebras, q-Heisenberg algebras and q-oscillator algebras are briefly described.     

On the quiver-theoretical quantum Yang–Baxter equation

Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum Yang–Baxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a “solution” for short. Results of Etingof–Schedler–Soloviev, Lu–Yan–Zhu and Takeuchi on the set-theoretical quantum Yang–Baxter equation are generalized to the context of quivers, with groupoids playing the role of groups. The notion of “braided groupoid” is introduced. Braided groupoids are solutions and are characterized in terms of bijective 1-cocycles. The structure groupoid of a non-degenerate solution is defined; it is shown that it is a braided groupoid. The reduced structure groupoid of a non-degenerate solution is also defined. Non-degenerate solutions are classified in terms of representations of matched pairs of groupoids. By linearization we construct star-triangular face models and realize them as modules over quasitriangular quantum groupoids introduced in papers by M. Aguiar, S. Natale and the author.     

Dual affine quantum groups

Let be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let be the dual Lie bialgebra. By dualizing the quantum double construction – via formal Hopf algebras – we construct a new quantum group , dual of . Studying its specializations at roots of 1 (in particular, its semi-classical limits), we prove that it yields quantizations of and (the formal Poisson group attached to ), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type. Received January 27, 1999     

A Littelmann path model for crystals of generalized Kac-Moody algebras

A Littelmann path model is constructed for crystals pertaining to a not necessarily symmetrizable Borcherds-Cartan matrix. Here one must overcome several combinatorial problems coming from the imaginary simple roots. The main results are an isomorphism theorem and a character formula of Borcherds-Kac-Weyl type for the crystals. In the symmetrizable case, the isomorphism theorem implies that the crystals constructed by this path model coincide with those of Jeong, Kang, Kashiwara and Shin obtained by taking q→0 limit in the quantized enveloping algebra.     

Representations of the Lie algebra of vector fields on a sphere

For an affine algebraic variety X we study a category of modules that admit compatible actions of both the algebra A of functions on X and the Lie algebra of vector fields on X. In particular, for the case when X is the sphere ${\mathbb{S}}^2$, we construct a set of simple modules that are finitely generated over A. In addition, we prove that the monoidal category that these modules generate is equivalent to the category of finite-dimensional rational ${\mathrm{GL}}_2$-modules.     

Unitarizability of premodular categories

We study the unitarizability of premodular categories constructed from representations of quantum group at roots of unity. We introduce Grothendieck unitarizability as a natural generalization of unitarizability to classes of premodular categories with a common Grothendieck semiring. We obtain new results for quantum groups of Lie types F₄ and G₂, and improve the previously obtained results for Lie types B and C.     

Measurable dynamics of maps on profinite groups

We study the measurable dynamics of transformations on profinite groups, in particular of those which factor through sufficiently many of the projection maps; these maps generalize the 1-Lipschitz maps on _p.