Nonstandard Poincare and Heisenberg Algebras

New deformations of the Poincare group Fun(P (1 + 1)) and its dual enveloping algebra U(p(1 + 1)) are obtained as a contraction of the hdeformed (Jordanian) quantum group Fun(SLh(2)) and its dual. A nonstandard quantization of the Heisenberg algebra U(h(1)) is also investigated. Ref. SISSA: 85/96/FM Off late, considerable interest has been generated towards the nonstandard quantization of Lie groups and algebras, commonly known as h or Jordanian deformation [1-4]. A peculiar feature is that the corresponding universal R matrix is triangular ie R = R21. Hence it is sometimes also called triangular deformation. These deformations were further extended to the case of supergroups [5].The contraction method is a useful technique to study inhomgeneous groups. This was employed first by Celeghini et al [11] to obtain quantization of some nonsemisimple groups. Recently attempts have been made to apply it to the Jordanian case [6-8] where the deformation paprameter h has a dimension , like the κ deformation [14]. In this letter we propose to obtain a nonstandard quantization of some of the simplest inhomogeneous groups the (1+1) dimensional Poincare group , its dual enveloping algebra, and the Heisenberg algebra U(h(1)) via a contraction of Fun(SLh(2)) and its dual Uh(sl(2)). The 3-dim Heisenberg algebra is further extended to 4 dimensions. Another deformation of the 2-dimensional Poincare group can be found in [8,9] which was obtained by simultaneously contracting the deformation parameter h. Our purpose here is to introduce a scaling of the generators in such a way that h remains unscaled. Fun(SLh(2, R)) is generated by the matrix T T = 