Higher level Zhu algebras and modules for vertex operator algebras

Motivated by the study of indecomposable, nonsimple modules for a vertex operator algebra V, we study the relationship between various types of V-modules and modules for the higher level Zhu algebras for V, denoted $A_n(V)$, for $n\in \mathbb{N}$, first introduced by Dong, Li, and Mason in 1998. We resolve some issues that arise in a few theorems previously presented when these algebras were first introduced, and give examples illustrating the need for certain modifications of the statements of those theorems. We establish that whether or not $A_{n?1}(V)$ is isomorphic to a direct summand of $A_n(V)$ affects the types of indecomposable V-modules which can be constructed by inducing from an $A_n(V)$-module, and in particular whether there are V-modules induced from $A_n(V)$-modules that were not already induced by $A_0(V)$. We give some characterizations of the V-modules that can be constructed from such inducings, in particular as regards their singular vectors. To illustrate these results, we discuss two examples of $A_1(V)$: when V is the vertex operator algebra associated to either the Heisenberg algebra or the Virasoro algebra. For these two examples, we show how the structure of $A_1(V)$ in relationship to $A_0(V)$ determines what types of indecomposable V-modules can be induced from a module for the level zero versus level one Zhu algebras. We construct a family of indecomposable modules for the Virasoro vertex operator algebra that are logarithmic modules and are not highest weight modules.     

On the defining relations for generalized q-Schur algebras

We show that the defining relations needed to describe a generalized q-Schur algebra as a quotient of a quantized enveloping algebra are determined completely by the defining ideal of a certain finite affine variety, the points of which correspond bijectively to the set of weights. This explains, unifies, and extends previous results.     

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.     

Hilbert Space Representations of Cross Product Algebras II

In this paper, we study and classify Hilbert space representations of cross product -algebras of the quantized enveloping algebra with the coordinate algebras of the quantum motion group and of the complex plane, and of the quantized enveloping algebra with the coordinate algebras of the quantum group and of the quantum disc. Invariant positive functionals and the corresponding Heisenberg representations are explicitly described.Presented by S.L. Woronowicz.     

Yangians: Their representations and characters

The finite-dimensional irreducible representations of the Yangian of sl₂ are parametrized by their highest weights, which are monic polynomials in one variable. In this paper, we give a formula for the character of such a representation which depends only on its highest weight, and is an analogue of the classical Weyl character formula.     

Higher Level Affine Crystals and Young Walls

Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals for quantum affine algebras of type , , , , , and . The irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls. The notion of slices and splitting of blocks plays a crucial role in the construction of crystals.Presented by Peter Littelman.     

On simple modules over conformal Galilei algebras

We study irreducible representations of two classes of conformal Galilei algebras in 1-spatial dimension. We construct a functor which transforms simple modules with nonzero central charge over the Heisenberg subalgebra into simple modules over the conformal Galilei algebras. This can be viewed as an analogue of oscillator representations. We use oscillator representations to describe the structure of simple highest weight modules over conformal Galilei algebras. We classify simple weight modules with finite dimensional weight spaces over finite dimensional Heisenberg algebras and use this classification and properties of oscillator representations to classify simple weight modules with finite dimensional weight spaces over conformal Galilei algebras.     

An equivalence of two constructions of permutation-twisted modules for lattice vertex operator algebras

The problem of constructing twisted modules for a vertex operator algebra and an automorphism has been solved in particular in two contexts. One of these two constructions is that initiated by the third author in the case of a lattice vertex operator algebra and an automorphism arising from an arbitrary lattice isometry. This construction, from a physical point of view, is related to the space–time geometry associated with the lattice in the sense of string theory. The other construction is due to the first author, jointly with C. Dong and G. Mason, in the case of a multifold tensor product of a given vertex operator algebra with itself and a permutation automorphism of the tensor factors. The latter construction is based on a certain change of variables in the worldsheet geometry in the sense of string theory. In the case of a lattice that is the orthogonal direct sum of copies of a given lattice, these two very different constructions can both be carried out, and must produce isomorphic twisted modules, by a theorem of the first author jointly with Dong and Mason. In this paper, we explicitly construct an isomorphism, thereby providing, from both mathematical and physical points of view, a direct link between space–time geometry and worldsheet geometry in this setting.     

Twisted Yangians and Mickelsson algebras I

We introduce an analogue of the composition of the Cherednik and Drinfeld functors for twisted Yangians. Our definition is based on the Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.     

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.     

Conformal nets associated with lattices and their orbifolds

In this paper, we study representations of conformal nets associated with positive definite even lattices and their orbifolds with respect to isometries of the lattices. Using previous general results on orbifolds, we give a list of all irreducible representations of the orbifolds, which generate a unitary modular tensor category.     

N-point locality for vertex operators: Normal ordered products,operator product expansions,twisted vertex algebras

In this paper we study fields satisfying N-point locality and their properties. We obtain residue formulae for N-point local fields in terms of derivatives of delta functions and Bell polynomials. We introduce the notion of the space of descendants of N-point local fields which includes normal ordered products and coefficients of operator product expansions. We show that examples of N  -point local fields include the vertex operators generating the boson–fermion correspondences of types B, C and D-A. We apply the normal ordered products of these vertex operators to the setting of the representation theory of the double-infinite rank Lie algebras _b∞$b_{\infty }$, _c∞$c_{\infty }$, _d∞$d_{\infty }$. Finally, we show that the field theory generated by N-point local fields and their descendants has a structure of a twisted vertex algebra.     

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.     

q-Difference intertwining operators and q-conformal invariant equations

We first recall a canonical procedure for the construction of the invariant differential operators and equations for arbitrary complex or real noncompact semisimple Lie groups. Then we present the application of this procedure to the case of quantum groups. In detail is given the construction of representations of the quantum algebra U _q (sl(n)) labelled by n–1 complex numbers and acting in the spaces of functions of n(n–1)/2 noncommuting variables, which generate a q-deformed SL(4) flag manifold. The conditions for reducibility of these representations and the procedure for the construction of the q-difference intertwining operators are given. Using these results for the case n=4 we propose infinite hierarchies of q-difference equations which are q-conformal invariant. The lowest member of one of these hierarchies are new q-Maxwell equations. We propose also new q-Minkowski spacetime which is part of a q-deformed SU(2,2) flag manifold.     

An analogue of the classical Yang-Baxter equation for general algebraic structures

In this paper, we introduce an analogue of the classical Yang-Baxter equation for general algebraic structures (including nonassociative algebras and vertex operator algebras). Moreover, we give several ways to construct solutions of the equation in case the algebraic structure is graded by an abelian group. In particular, we construct some unitary nondegenerate trignometric solutions of the classical Yang-Baxter equation for affine Lie algebras by means of our equation.This paper was written while the author was a graduate student in the Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903, U.S.A.     

Elliptic Lie algebras and tubular algebras

This article is to study relations between tubular algebras of Ringel and elliptic Lie algebras in the sense of Saito-Yoshii. Using the explicit structure of the derived categories of tubular algebras given by Happel-Ringel, we prove that the elliptic Lie algebra of type , , or is isomorphic to the Ringel-Hall Lie algebra of the root category of the tubular algebra with the same type. As a by-product of our proof, we obtain a Chevalley basis of the elliptic Lie algebra following indecomposable objects of the root category of the corresponding tubular algebra. This can be viewed as an analogue of the Frenkel-Malkin-Vybornov theorem in which they described a Chevalley basis for each untwisted affine Kac-Moody Lie algebra by using indecomposable representations of the corresponding affine quiver.     

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.     

Endomorphism algebras and q-traces

For a braided vector space (V,σ) with braiding σ of Hecke type, we introduce three associative algebra structures on the space of graded endomorphisms of the quantum symmetric algebra S_σ(V). We use the second product to construct a new trace. This trace is an algebra morphism with respect to the third product. In particular, when V is the fundamental representation of U_qsl_N+1 and σ is the action of the R-matrix, this trace is a scalar multiple of the quantum trace of type A.     

Quantization of formal classical dynamical r-matrices: the reductive case

In this paper we prove the existence of a formal dynamical twist quantization for any triangular and non-modified formal classical dynamical r-matrix in the reductive case. The dynamical twist is constructed as the image of the dynamical r-matrix by a L_∞-quasi-isomorphism. This quasi-isomorphism also allows us to classify formal dynamical twist quantizations up to gauge equivalence.