Department of Mathematics,
Chalmers University of Technology
Dept of Math, Chalmers
S-412 96 Goteborg, Sweden

E-mail: astolin@math.chalmers.se

q-Deformed Power Function over q-Commuting Variables and Drinfeld's Problem of Quantization of Lie Bialgebras

Abstract:
We find certain functional identities for the Gauss q-power function of a sum of q-commuting variables. Then we use these identities to obtain two-parameter twists of the quantum affine algebra U_q (\widehat{sl}_2) and of the Yangian Y(sl_2). We determine the corresponding deformed trigonometric and rational quantum R-matrices, which then are used in the computation of deformed XXX and XXZ Hamiltonians. It turns out that these deformed trigonometric and rational R-matrices quantize classical trigonometric and rational r-matrices in sl_2 found by Belavin, Drinfeld (trigonometric) and Stolin (rational). Therefore in the sl_2 case  we answer the following problem posed by Drinfeld: find an explicit quantization of non-standard trigonometric r-matrices.