Quantum Algebra And Topology

We show the existence of a universal Vassiliev invariant for links in closed surface cylinders by explicit construction using configuration space integrals.

Two families (type A and type B ) of confluent hypergeometric polynomials in several variables are studied. We describe the orthogonality properties, differential equations, and Pieri type recurrence formulas for these families. In the one-variable case, the polynomials in question reduce to the Hermite polynomials (type A ) and the Laguerre polynomials (type B ), respectively. The multivariable confluent hypergeometric families considered here may be used to diagonalize the rational quantum Calogero models with harmonic confinement (for the classical root systems) and are closely connected to the (symmetric) generalized spherical harmonics investigated by Dunkl.

In this paper we study a class of modules over infinite-dimensional Lie (super)algebras, which we call conformal modules. In particular we classify and construct explicitly all irreducible conformal modules over the Virasoro and the N=1 Neveu-Schwarz algebras, and over the current algebras.

A rational Ansatz is proposed for the generating function ∑ j,k β 2j+k,2j x j y k , where β m,u is the number of primitive chinese character diagrams with u univalent and 2m−u trivalent vertices. For P m := ∑ u≥2 β m,u , the conjecture leads to the sequence 1,1,1,2,3,5,8,12,18,27,39,55, 78,108,150,207,284,388,532,726 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – for primitive chord diagrams of degrees m≤20 , with predictions underlined. The asymptotic behaviour lim m→∞ P m / r m =1.06260548918755 results, with r=1.38027756909761 solving r 4 = r 3 +1 . Vassiliev invariants of knots are then enumerated by 0,1,1,3,4,9,14,27,44,80,132,232, 384,659,1095,1851,3065,5128,8461,14031 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – and Vassiliev invariants of framed knots by 1,2,3,6,10,19,33,60,104,184,316,548, 932,1591,2686,4537,7602,12730,21191,35222 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – These conjectures are motivated by successful enumerations of irreducible Euler sums. Predictions for β 15,10 , β 16,12 and β 19,16 suggest that the action of sl and osp Lie algebras, on baguette diagrams with ladder insertions, fails to detect an invariant in each case.

We show that there is an operator with a simple geometric significance which yields the ordinary geometry of a linear equidistant lattice via Connes' distance function.

Lie algebra contractions on the classical Drinfel'd Double of a given Lie bialgebra are introduced and compared to the usual Lie bialgebra contraction theory. The connection between both approaches turns out to be intimately linked to duality problems. The non-relativistic (Galilean) limit of a (1+1) Poincaré Double is used to illustrate the contraction process. Finally, it is shown that, in a certain sense, the classical limit in a quantum algebra can be thought as a certain contraction on the corresponding Double.

The interpretation of the Meixner-Pollaczek, Meixner and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra su(1,1) and the Clebsch-Gordan decomposition leads to generalisations of the convolution identities for these polynomials. Using the Racah coefficients convolution identities for continuous Hahn, Hahn and Jacobi polynomials are obtained. From the quantised universal enveloping algebra for su(1,1) convolution identities for the Al-Salam and Chihara polynomials and the Askey-Wilson polynomials are derived by using the Clebsch-Gordan and Racah coefficients. For the quantised universal enveloping algebra for su(2) q-Racah polynomials are interpreted as Clebsch-Gordan coefficients, and the linearisation coefficients for a two-parameter family of Askey-Wilson polynomials are derived.

Let B be the crystal basis of the minus part of the quantized enveloping algebra of a semi-simple Lie algebra. Kashiwara has shown that B has a combinatorial description in terms of an embedding of B into the tensor product of B and k abstract crystals B_{i_j}, j=1,2,...,k, where the longest word in the Weyl group is s_{i_1}...s_{i_k}. We give an explicit description of the image of this embedding for classical Lie algebras of types A, B, C, D. This description is in terms of semi-standard Young tableaux of types A, B, C, D defined by Kashiwara and Nakashima.

We study, in the path realization, crystals for Demazure modules of affine Lie algebras of types A (1) n , B (1) n , C (1) n , D (1) n , A (2) 2n−1 , A (2) 2n ,and D (2) n+1 . We find a special sequence of affine Weyl group elements for the selected perfect crystal, and show if the highest weight is $l\La_0$, the Demazure crystal has a remarkably simple structure.

The Batalin-Vilkovisky master equations, both classical and quantum, are precisely the integrability equations for deformations of algebras and differential algebras respectively. This is not a coincidence; the Batalin-Vilkovisky approach is here translated into the language of deformation theory.