Special Solutions of the Quantum Yang-Baxter Equation

We present solutions of the Quantum Yang-Baxter Equation that satisfy the condition R ab cd 6= 0 ) (fa; bg = fc; dg) or (b = oe(a) and d = oe(c)); where oe denotes the involution on f1; : : : ; ng given by oe(i) = n + 1 \Gamma i. AMS Subject Classification (1991): 81R50, 57M25. Keywords and Phrases: multiparameter R-matrix, Quantum Yang-Baxter equation. Note: The author is supported by NWO, Grant N. 611-307-100. 1 Introduction In this report we construct special solutions of the Quantum Yang-Baxter Equation (QYBE). The QYBE involves a regular n 2 \Theta n 2 -matrix R over the field of complex numbers and can shortly be written as R12R13R23 = R23R13R12 . In this equation, R12 denotes the n 3 \Theta n 3 - matrix that arises by letting R act on the first and second factor of the tensor product C n\Omega C n\Omega C n . The matrices R13 and R23 are defined similarly. Written out in components the QYBE takes the following form: X i;j;k R ab ij R ic uk R jk vw = ..

From exponential coordinates to bicovariant differential calculi on matrix quantum groups

A procedure to obtain bicovariant differential calculi on matrix quantum groups is presented. The construction is based on the description of the matrix quantum group as a quantized universal enveloping algebra by the use of exponential coordinates. The procedure is illustrated by applying it to the two-dimensional solvable quantum group and the Heisenberg quantum group

A natural differential calculus on Lie bialgebras with dual of triangular type

We prove that for a specific class of Lie bialgebras, there exists a natural differential calculus. This class consists of the Lie bialgebras for which the dual Lie bialgebra is of triangular type. The differential calculus is explicitly constructed with the help of the $R$-matrix from the dual. The method is illustrated by several examples

Quantization of differential calculi on universal enveloping algebras

A method to construct differential calculi on quantized universal enveloping algebras is discussed. These differential calculi are obtained by quantizing calculi on 'classical' enveloping algebras provided with appropriate co‐Poisson structures. The procedure is demonstrated by applying it to the standard quantizations of the Heisenberg algebra and the algebragl(2)