ar X iv : q - a lg / 9 50 50 06 v 1 4 M ay 1 99 5 SPBU - IP - 95 - 8 Recursion relations and branching rules for simple Lie algebras

ar X iv : q - a lg / 9 70 50 22 v 1 2 7 M ay 1 99 7 Coloured Hopf

Some new algebraic structures related to the coloured Yang-Baxter equation, and termed coloured Hopf algebras, are reviewed. Coloured quantum universal enveloping algebras of Lie algebras are defined in this context. An extension to the coloured graded Yang-Baxter equation and coloured Hopf superalgebras is also presented. The coloured two-parameter quantum universal enveloping algebra of gl(1/...

ar X iv : q - a lg / 9 70 50 27 v 1 2 8 M ay 1 99 7 Jordanian U h , s gl ( 2 ) and its coloured realization

A two-parametric non-standard (Jordanian) deformation of the Lie algebra gl(2) is constructed, and then, exploited to obtain a new, triangular R-matrix solution of the coloured Yang-Baxter equation. The corresponding coloured quantum group is presented explicitly.

ar X iv : q - a lg / 9 50 30 07 v 1 1 6 M ar 1 99 5 Field Theory on q = − 1 Quantum Plane

We build the q = −1 defomation of plane on a product of two copies of algebras of functions on the plane. This algebra constains a subalgebra of functions on the plane. We present general scheme (which could be used as well to construct quaternion from pairs of complex numbers) and we use it to derive differential structures, metric and discuss sample field theoretical models. TPJU 4/95 Februar...

ar X iv : a lg - g eo m / 9 50 20 26 v 2 9 M ay 1 99 5 ALGEBRAIC SURFACES AND SEIBERG - WITTEN INVARIANTS

ar X iv : q - a lg / 9 70 50 12 v 1 1 6 M ay 1 99 7 Poisson structures on the center

It is shown that the elliptic algebra Aq,p(ŝl(2)c) has a non trivial center at the critical level c = −2, generalizing the result of Reshetikhin and Semenov-Tian-Shansky for trigonometric algebras. A family of Poisson structures indexed by a non-negative integer k is constructed on this center.