Quantum Algebra And Topology

A classical theorem of Scheunert on G -color Lie algebras, asserts in the case of finitely generated abelian groups, one can twist the algebra structure and the commutation bicharacter on G by a 2-cocycle twist to a super-Lie G graded, algebra. In this paper we show that this can be done for an arbitrary group.

We produce a master identity {m}{m}=0 for homotopy Gerstenhaber algebras, as defined by Getzler and Jones and utilized by Kimura, Voronov, and Zuckerman in the context of topological conformal field theories. To this end, we introduce the notion of a "partitioned multilinear map" and explain the mechanics of composing such maps. In addition, many new examples of pre-Lie algebras and homotopy Gerstenhaber algebras are given.

We give a new construction of the moonshine VOA V^{\natural} over the real number field. We proved that V^{\natural} has a positive definite invariant bilinear form and its full automorphism group is the Monster simple group. We also construct an infinite series of meromorphic VOAs whose full automorphism groups are finite. We calculate the trace form on V^{\natural} for some element of the Monster.

Symmetric Jack polynomials arise naturally in several contexts, including statistics, physics, combinatorics, and representation theory. They are pairwise orthogonal with repsect to two different inner products, the first defined by integration over an n-dimensional torus, and the second defined via a power series expansion. The nonsymmetric analogs of Jack polynomials were recently defined by Opdam via the first inner product. In this paper we show that they are also orthogonal with respect to an analog of the second product which was proposed by Dunkl. We also derive an explict formula for their norms.

Nonlinear deformations of the enveloping algebra of su(2), involving two arbitrary functions of J_0 and generalizing the Witten algebra, were introduced some time ago by Delbecq and Quesne. In the present paper, the problem of endowing some of them with a Hopf algebraic structure is addressed by studying in detail a specific example, referred to as A + q (1) . This algebra is shown to possess two series of (N+1)-dimensional unitary irreducible representations, where N=0, 1, 2, .... To allow the coupling of any two such representations, a generalization of the standard Hopf axioms is proposed by proceeding in two steps. In the first one, a variant and extension of the deforming functional technique is introduced: variant because a map between two deformed algebras, su_q(2) and A + q (1) , is considered instead of a map between a Lie algebra and a deformed one, and extension because use is made of a two-valued functional, whose inverse is singular. As a result, the Hopf structure of su_q(2) is carried over to A + q (1) , thereby endowing the latter with a double Hopf structure. In the second step, the definition of the coproduct, counit, antipode, and R-matrix is extended so that the double Hopf algebra is enlarged into a new algebraic structure. The latter is referred to as a two-colour quasitriangular Hopf algebra because the corresponding R-matrix is a solution of the coloured Yang-Baxter equation, where the `colour' parameters take two discrete values associated with the two series of finite-dimensional representations.

We show that the bases of irreducible integrable highest weight module of a non-symmetric Kac-Moody algebra, which is associated to a quiver with a nontrivial admissible automorphism, can be naturally identified with a set of certain invariant Langrangian irreducible subvarieties of certain varieties associated with the quiver defined by Nakajima. In the case of non-symmetric affine or finite Kac-Moody algebras, the bases can be naturally identified with a set of certain invariant Langrangian irreducible subvarieties of a particular deformation of singularities of the moduli space of instantons over A-L-E spaces. The motivation of this paper comes from string/string duality and the paper is ended with questions and speculations related to string/string duality.

In this paper we derive from arguments of string scattering a set of eight tetrahedron equations, with different index orderings. It is argued that this system of equations is the proper system that represents integrable structures in three dimensions generalising the Yang-Baxter equation. Under additional restrictions this system reduces to the usual tetrahedron equation in the vertex form. Most known solutions fall under this class, but it is by no means necessary. Comparison is made with the work on braided monoidal 2-categories also leading to eight tetrahedron equations.

Recently, Andrews and Berkovich introduced a trinomial version of Bailey's lemma. In this note we show that each ordinary Bailey pair gives rise to a trinomial Bailey pair. This largely widens the applicability of the trinomial Bailey lemma and proves some of the identities proposed by Andrews and Berkovich.

We use factorized L operator to construct an integrable model with open boundary conditions. By taking trigonometic limit( τ→ −1 − − − √ ∞ ) and scaling limit( ω→0 ), we get a Hamiltonian of a classical integrable system. It shows that this integrable system is similar to those found by Calogero et al.

The paper is devoted to the proof of the following conjecture due to B. Feigin. Let u ℓ be the small quantum group a the primitive ℓ -th root of unity. Then it is known that the usual Ext algebra of the trivial u ℓ -module is isomorphic to the algebra of regular functions on the nilpotent variety N in the corresponding simple Lie algebra g (see [GK]). Consider semiinfinite cohomology of the trivial u ℓ -module introduced in [Ar1], [Ar2]. It was shown in [Ar2] that the Ext algebra of the trivial u ℓ -module acts naturally on semiinfinite cohomology. Moreover semiinfinite cohomology space of the trivial ℓ -module is equipped with a natural g -module structure. B. Feigin conjectured that the described g -module and F(N) -module structures coincide with the ones on the space of local cohomology of the structure sheaf on N with support in the standard positive nilpotent subalgebra n + ⊂N⊂g . We give a detailed proof of the conjecture. Moreover we generalize the statement and describe semiinfinite cohomology of contragradient Weyl modules in terms of local cohomology of certain coherent sheaves on the nilpotent cone and on its desingularization T ∗ (G/B) . To do this we construct a certain specialization of the quantum BGG resolution of the simple module L(λ) defined for generic values of the quantizing parameter into the root of unity called the quasi-BGG complex. The complex conists of direct sums of quasi-Verma modules and provides conjecturally a resolution of the Weyl module W(λ) .