Quantum Algebra And Topology

The product of two Heisenberg-Weil algebras contains the Jordan-Schwinger representation of su(2). This Algebra is quotiented by the square-root of the Casimir to produce a non-associative algebra denoted by Ψ . This algebra may be viewed as the right-module over one of its associative subalgebras which corresponds to the algebra of scalar fields on the noncommutative sphere. It is now possible to interpret other subspaces as the space of spinor or vector fields on the noncommutative sphere. A natural basis of Ψ is given which may be interpreted as the deformed entries in the rotation matrices of SU(2).

An infinite dimensional algebra which is a non-decomposable reducible representation of su(2) is given. This algebra is defined with respect to two real parameters. If one of these parameters is zero the algebra is the commutative algebra of functions on the sphere, otherwise it is a noncommutative analogue. This is an extension of the algebra normally refered to as the (Berezin) quantum sphere or ``fuzzy'' sphere. A natural indefinite ``inner'' product and a basis of the algebra orthogonal with respect to it are given. The basis elements are homogenious polynomials, eigenvectors of a Laplacian, and related to the Hahn polynomials. It is shown that these elements tend to the spherical harmonics for the sphere. A Moyal bracket is constructed and shown to be the standard Moyal bracket for the sphere.

The relationship between the exactness of a first order differential calculus on a comodule algebra P and the Galois property of P is investigated.

Let g be an untwisted affine Kac-Moody algebra over the field K , and let U q (g) be the associated quantum enveloping algebra; let U q (g) be the Lusztig's integer form of U q (g) , generated by q -divided powers of Chevalley generators over a suitable subring R of K(q) . We prove a Poincaré-Birkhoff-Witt like theorem for U q (g) , yielding a basis over R made of ordered products of q -divided powers of suitable quantum root vectors.

A proof of Poincaré-Birkhoff-Witt theorem is given for a class of generalized Lie algebras closely related to the Gurevich S-Lie algebras. As concrete examples, we construct the positive (negative) parts of the quantized universal enveloping algebras of type A_n and M_{p,q,e}(n,K), which is a non-standard quantum deformation of GL(n). In particular, we get, for both algebras, a unified proof of the Poincaré-Birkhoff-Witt theorem and we show that they are genuine universal enveloping algebras of certain generalized Lie algebras.

We explicitly construct a star product for the complex Grassmann manifolds using the method of phase space reduction. Functions on C (p+q)⋅p ∗ , the space of (p+q)×p matrices of rank p, invariant under the right action of Gl(p,C) can be regarded as functions on the Grassmann manifold G p,q (C) , but do not form a subalgebra whereas functions only invariant under the unitary subgroup U(p)⊂Gl(p,C) do. The idea is to construct a projection from U(p) - onto Gl(p,C) -invariant functions, whose kernel is an ideal. This projection can be used to define a star-algebra on G p,q (C) onto which this projection acts as an algebra-epimorphism.

We propose and prove a trinomial version of the celebrated Bailey's lemma. As an application we obtain new fermionic representations for characters of some unitary as well as nonunitary models of N = 2 superconformal field theory (SCFT). We also establish interesting relations between N = 1 and N = 2 models of SCFT with central charges (3/2)(1−2(2−4ν ) 2 /2(4ν)) and 3(1−2/4ν) . A number of new mock theta function identities are derived.

In this elementary paper we prove that the extra vanishing property characterizes the BC interpolation Macdonald polynomials inside a very general class of multivariate interpolation polynomials. It follows that they are the only polynomials in this class that admit a tableaux sum formula or a q-integral representation.

We construct a Hopf algebra cocycle in the Yangian double DY(S L 2 ) , conjugating Drinfeld's coproduct to the usual one. To do that, we factorize the twist between two ``opposite'' versions of Drinfeld's coproduct, introduced in earlier work by V. Rubtsov and the first author, using the decomposition of the algebra in its negative and non-negative modes subalgebras.

An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra U q ( sl ^ 2 ) . A similar construction is proposed for the elliptic algebra A q,p ( sl ^ 2 ) .