h-deformation of Gr(2)

The h-deformation of functions on the Grassmann matrix group Gr(2) is presented via a contraction of Grq(2). As an interesting point, we have seen that, in the case of the h-deformation, both R-matrices of GLh(2) and Grh(2) are the same. E-mail: [email protected] E-mail: [email protected] In recent years a new class of quantum deformations of Lie groups and algebras, the so-called h-deformation, has been intensively studied by many authors [1-9]. The h-deformation of matrix groups can be obtained using a contraction procedure. We start with a quantum plane and its dual and follow the contraction method of [9]. Consider the q-deformed algebra of functions on the quantum plane [10] generated by x, y with the commutation rule x′y′ = qy′x′. (1) Applying a change of basis in the coordinates of the (1) by use of the following matrix g = ( 1 f 0 1 ) f = h q − 1 (2) one arrives at [9], in the limit q → 1, xy = yx+ hy. (3) We denote the quantum h-plane by Rh(2). Similarly, one gets the dual quantum h-plane R h(2) as generated by η, ξ with the relations ξ = 0 η = hηξ ηξ + ξη = 0. (4)