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Paper IPM / P / 6485  


Abstract:  
It is shown that the finite dimensional ireducible representations of the quantum matrix algebra M_{q}(3) ( the coordinate ring of GL_{q}(3) ) exist only when q is a root of unity ( q^{p} = 1 ). The dimensions of these representations can only be one of the following values: p^{3} , [( p^{3})/2 ] , [( p^{3})/4 ] or [( p^{3})/8 ] . The topology of the space of states ranges between two extremes , from a 3dimensional torus S^{1} ×S^{1} ×S^{1} ( which may be thought of as a generalization of the cyclic representation ) to a 3dimensional cube [ 0 , 1 ]×[ 0 , 1 ]×[ 0 , 1 ] .
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