Quantum Algebra And Topology

The paper contains essentially two new results. Physically, a deformation of the parastatistics in a sense of quantum groups is carried out. Mathematically, an alternative to the Chevalley description of the quantum orthosymplectic superalgebra U_q[osp(2n+1/2m)] in terms of m pairs of deformed parabosons and n pairs of deformed parafermions is outlined.

By using the generating function formula for the product of two q-Hermite polynomials q-deformation of the Feynman Green function for the harmonic oscillator is obtained.

We give a skein theoretic proof the Reshetikhin hook length formula for quantum dimension for the quantum group U_q(sl(N)).

We construct a solution of Cherednik's quantum Knizhnik Zamolodchikov equation associated with the root system of type C n . This solution is given in terms of a restriction of a q -Jordan-Pochhammer integral. As its applicaton, we give an explicit expression of a special case of the Macdonald polynomial of the C n type. Finally we explain the connection with the representation of the Hecke algebra.

We apply the ideas of Alvarez and Labastida to the invariant of spin networks defined by Witten and Martin based on Chern-Simons theory. We show that it is possible to define ambient invariants of spin networks that (for the case of SU(2)) can be considered as extensions to spin networks of the Jones polynomial. Expansions of the coefficients of the polynomial yield primitive Vassiliev invariants. The resulting invariants are candidates for solutions of the Wheeler--DeWitt equations in the spin network representation of quantum gravity.

We define a three-parameter deformation of the Weyl-Heisenberg algebra that generalizes the q-oscillator algebra. By a purely algebraical procedure, we set up on this quantum space two differential calculi that are shown to be invariant on the same quantum group, extended to a ten-generator Hopf-star-algebra. We prove that, when the values of the parameters are related, the two differential calculi reduce to one that is invariant under two quantum groups.

Categorial actions of braided tensor categories are defined and shown to be the right framework for a discussion of the categorial structure related to the group of braids in the cylinder. A Kauffman polynomial of links in the solid torus is constructed.

We use an explicit isomorphism from the representation ring of the quantum group U_q(sl(N)) to the Homfly skein of the annulus, to determine an element of the skein which is the image of the mth Adams operator, \psi_m, on the fundamental representation, c_1. This element is a linear combination of m very simple m-string braids. Using this skein element, we show that the Vassiliev invariant of degree n in the power series expansion of the U_q(sl(N)) quantum invariant of a knot coloured by \psi_m(c_1) is the canonical Vassiliev invariant with weight system W_n\psi_m^{(n)} where W_n is the weight system for the Vassiliev invariant of degree n in the expansion of the quantum invariant of the knot coloured by c_1 and \psi_m^{(n)} is the Adams operator on n-chord diagrams defined by Bar-Natan.

We study a q -deformation for the semi-direct product of the symmetric group S n with the Clifford algebra on n anticommuting generators. We obtain a q -version of the projective analogue for the classical Young symmetrizer found by the second author [Adv. Math. 127(1997), 190-257]. Our main tool is an analogue of the Hecke algebra of complex valued functions on the group G L n over a p -adic field relative to the Iwahori subgroup.

We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed W n algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the deformed W 3 algebra.