ar X iv : q - a lg / 9 60 70 15 v 1 1 4 Ju l 1 99 6 Flag Varieties and the Yang - Baxter Equation

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 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 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/...

X iv : q - a lg / 9 60 20 01 v 1 1 F eb 1 99 6 Poisson structures on the Poincaré group

An introduction to inhomogeneous Poisson groups is given. Poisson inhomogeneous O(p, q) are shown to be coboundary, the generalized classical Yang-Baxter equation having only one-dimensional right hand side. Normal forms of the classical r-matrices for the Poincaré group (inhomogeneous O(1, 3)) are calculated.

ar X iv : h ep - t h / 95 07 12 3 v 1 2 4 Ju l 1 99 5 May 1995 Simple Construction of Elliptic Boundary K - matrix ¶

We give the infinite-dimensional representation for the elliptic K-operator satisfying the boundary Yang-Baxter equation. By restricting the functional space to finite-dimensional space, we construct the elliptic K-matrix associated to Belavin's completely Z-symmetric R-matrix.