Quantization of tensor representations and deformation of matrix bialgebras

Abstract

The quantum matrix bialgebra M q (2) and quantum plane k 2 q are constructed as preferred deformations of the classical matrix bialgebra and plane, that is, the comultiplication for M q (2) and the M q (2)-coaction for k 2 q remain unchanged on all elements (not just generators) during the deformation. The construction of these algebras is obtained by quantizing the standard representations of the Lie algebra ▪(2) and the appropriate symmetric group on each tensor power of the vector space of coordinate functions on the plane. Analyzing the invariant elements of these representations then leads to the desired deformations.