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 constraint for $R$ or the braid equation for $\hat{R}=PR$. For three biparametric deformatios, $G{L}_{(p,q)}(2),G{L}_{(g,h)}(2)$ and $G{L}_{(q,h)}(1/1)$, the standard,the nonstandard and the hybrid one respectively, $R$ or $\hat{R}$ is found to depend, apart from the two parameters defining the deformation in question, on an extra free parameter $K$,such that only for two values of $K$, given explicitly for each case, 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 Schak. Various properties of the triparametric $\hat{R}(K;p,q)$, $\hat{R}(K;g,h)$ and $\hat{R}(K;q,h)$ are studied. In the larger space of the modified braid equation (MBE) even $\hat{R}(K;p,q)$ can satisfy ${\hat{R}}^2=1$ outside braid equation (BE) subspace. A generalized, $K$- dependent, Hecke condition is satisfied by each 3-parameter $\hat{R}$. 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.

Latex, 17 pages, minor changes

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Cited by in corpus (3)

• Spectral Decomposition and Baxterisation of Exotic Bialgebras and Associated Noncommutative Geometries
• Modified braid equations, Baxterizations and noncommutative spaces for the quantum groups $G{L}_q(N),S{O}_q(N)$ and $S{p}_q(N)$
• Modified Braid Equations for SO_q (3) and noncommutative spaces