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RTT relations, a modified braid equation and noncommutative planes
With the known group relations for the elements of a quantum
matrix as input a general solution of the relations is sought without
imposing the Yang - Baxter constraint for or the braid equation for
. For three biparametric deformatios, and , the standard,the nonstandard and the
hybrid one respectively, or is found to depend, apart from the
two parameters defining the deformation in question, on an extra free parameter
,such that only for two values of , given explicitly for each case, one
has the braid equation. Arbitray corresponds to a class (conserving the
group relations independent of ) of the MQYBE or modified quantum YB
equations studied by Gerstenhaber, Giaquinto and Schak. Various properties of
the triparametric , and are
studied. In the larger space of the modified braid equation (MBE) even
can satisfy outside braid equation (BE)
subspace. A generalized, - dependent, Hecke condition is satisfied by each
3-parameter . The role of in noncommutative geometries of the
, and 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.Comment: Latex, 17 pages, minor change
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