ar X iv : q - a lg / 9 70 20 09 v 1 6 F eb 1 99 7 The Fundamental Theorem of Vassiliev Invariants

ar X iv : q - a lg / 9 60 70 28 v 1 2 7 Ju l 1 99 6 Examples of Categorification

ar X iv : q - a lg / 9 70 30 25 v 3 2 6 A pr 1 99 8 WHEELS , WHEELING , AND THE KONTSEVICH INTEGRAL OF THE UNKNOT

We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne [De]) for the relation between the two natural products on the space of Chinese characters. The two formulas use the related notions of “Wheels” and “Wheeling”. We prove these formulas ‘on the level of Lie algebras’ using standard techniques from the...

ar X iv : q - a lg / 9 70 90 01 v 1 1 S ep 1 99 7 Bilinear identity for q - hypergeometric integrals

ar X iv : q - a lg / 9 70 10 08 v 2 4 F eb 1 99 7 FACTORIZATION OF DIFFERENTIAL OPERATORS , QUASIDETERMINANTS , AND NONABELIAN TODA FIELD EQUATIONS

We integrate nonabelian Toda field equations [Kr] for root systems of types A, B, C, for functions with values in any associative algebra. The solution is expressed via quasideterminants introduced in [GR1],[GR2], [GR4]. In the appendix we review some results concerning noncommutative versions of other classical integrable equations.

ar X iv : q - a lg / 9 70 50 20 v 1 2 7 M ay 1 99 7 QUANTUM GROUP DUALITY AND THE CUNTZ ALGEBRA

The Cuntz algebra carries in a natural way the structure of a module algebra over the quantized universal enveloping algebra Uq(g), and the structure of a co-module algebra over the quantum group Gq associated with Uq(g). These two algebraic structures are dual to each other via the duality between Gq and Uq(g).