M ar 2 00 1 Modified Braid Equations for SO q ( 3 ) and noncommutative spaces

General solutions of thê RT T equation with a maximal number of free parameters in the specrtal decomposition of vector SO q (3) ˆ R matrices are implemented to construct modified braid equations (M BE). These matrices conserve the given, standard, group relations of the nine elements of T , but are not constrained to satisfy the standard braid equation (BE). Apart from q and a normalisation factor ourˆR contains two free parameters, instead of only one such parameter for deformed unitary algebras studied in a previous paper [1] where the nonzero right hand side of the M BE had a linear term proprotional to (ˆ R (12) − ˆ R (23)). In the present case the r.h.s. is, in general, nonliear. Several particular solutions are given (Sec.2) and the general structure is analysed (App.A). Our formulation of the problem in terms of projectors yield also two new solutions of standard (nonmodified) braid equation (Sec.2) which are further discussed (App.B). The noncommutative 3-spaces obtained by implementing such generalizedˆR matrices are studied (Sec.3). The role of coboundaryˆR matrices (not satisfying the standard BE) is explored. The M BE and Baxterization are presented as complementary facets of the same basic construction, namely, the general solution ofˆRT T equation (Sec.4). A new solution is presented in this context. As a simple but remarkable particular case a nontrivial solution of BE is obtained (App.B) for q = 1. This solution has no free parameter and is not obtainable by twisting the identity matrix. In the concluding remarks (Sec.5), among other points, generalisation of our results to SO q (N) is discussed.