Rational-trigonometric deformation

Abstract

We discuss a rational-trigonometric deformation for two algebraic cases; for the universal enveloping algebra U(g[u]) of a polynomial loop algebra g[u], where g is a finite-dimensional complex simple Lie algebra, and for the two-dimensional plane (x, y). In the both cases these deformations are obtained by a singular transformation (at q=1) of the q-deformation of U(g[u]) and (x, y). In the first case the quantum Hopf algebra called Drinfeldian Dqη(g) is a quantization of U(g[u]) in the direction of a classical r-matrix which is a sum of the simple rational and trigonometric r-matrices. The Drinfeldian Dqη(g) contains Uq(g) as a Hopf subalgebra, moreover Uq(g[u]) and Yangian Yη(g) are its limit quantum algebras when the deformation parameters η goes to 0 and q goes to 1, respectively. Using the rational-trigonometric deformation of the plane (x, y) we introduce the (q,η)- and η-numbers, (q,η)- and η-exponentials, and (q,η)- and η-hypergeometric series.