RTT relations, a modified braid equation and noncommutative planes

Abstract

With the known group relations for the elements (a,b,c,d) of a quantum matrix T as input a general solution of the RTT relations is sought without imposing the Yang–Baxter (YB) constraint for R or the braid equation for R̂=PR. For three biparametric deformatios, GL(p,q)(2), GL(g,h)(2), and GL(q,h)(1/1), the standard, the nonstandard, and the hybrid one, respectively, R or R̂ is found to depend, apart from the two parameters defining the deformation in question, on an extra free parameter K, such that R̂(12)R̂(23)R̂(12)−R̂(23)R̂(12)R̂(23)=[(K/K1)−1][(K/K2)−1](R̂(23)−R̂(12)) with (K1,K2)=(1,p/q),(1,1), and (1,1/q), respectively. Only for K=K1 or K=K2 one has the braid equation. Arbitray K corresponds to a class (conserving the group relations independent of K) of the MQYBE or modified quantum YB equations studied by Gerstenhaber, Giaquinto, and Schack. Various properties of the triparametric R̂(K;p,q), R̂(K;g,h), and R̂(K;q,h) are studied. In the larger space of the modified braid equation (MBE) even R̂(K;p,q) can satisfy R̂2=1 outside the braid equation (BE) subspace. A generalized, K-dependent, Hecke condition is satisfied by each three-parameter R̂. The role of K in noncommutative geometries of the (K;p,q), (K;g,h), and (K;q,h) deformed planes is studied. K is found to introduce a “soft symmetry breaking,” preserving most interesting properties and leading to new interesting ones. Further aspects to be explored are indicated.