On the Inhomogeneous Quantum Groups

Using the multi-parametric deformation of the algebra of functions on GL(n + 1) and the universal enveloping algebra U (igl(n+ 1)), we construct the multi-parametric quantum groups IGLq(n) and Uq (igl(n)). The quantization of inhomogeneous groups is not well stablished. These non-semisimple groups are important, both from a purely mathematical point of view , and also because one of the most important groups appearing in physics, viz. the Poincaré group, is inhomogeneous. Various authors in the last few years studied the quantization of these objects from different viewpoints and by different methods [1-7]. In this paper we consider the inhomogeneous general linear group IGL(n). We will show that in analogy with the classical case, the multi-parameteric deformation of the parent GL(n+1) will lead to a consistent quantization of both the algebra of functions and the universal enveloping algebra of the inhomogeneous group IGL(n). Multi-parametric quantum groups have been studied by various authors [8-14]. In this paper we will use the notations of Ref. [13]. To clarify our approach, we will first consider the two dimensional case IGL(2); then we will generalize to the higher dimensions. Throughout this paper, when we use a subscript q for an object, we mean the multi-parameteric deformation of that object. The number of, and the conditions on, the parameters will be clear from the context. Let us consider the two dimensional inhomogeneous group IGL(2). Its action on the two di1 mensional plane IR is: x 7−→ x = ax+ by + u y 7−→ y = cx+ dy + v. (1) This action may be represented on the z = 1 subset of IR by means of the following matrix: