Representations of The Coordinate Ring of $ GL_{q}(n) $}

Details Representations of The Coordinate Ring of $ GL_{q}(n) $} Representations of The Coordinate Ring of $ GL_{q}(n) $}

1993-06-12

arxiv.org/abs/hep-th/9306058v1

Physics

High Energy Physics

High Energy Physics - Theory

20 pages ,report #. 93-020

Scientific paper

It is shown that the finite dimensional irreducible representations of the quantum matrix algebra $ M_q(n) $ ( the coordinate ring of $ GL_q(n) $ ) 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^N \over 2^k } $ where $ N = {n(n-1)\over 2 } $ and $ k \in \{ 0, 1, 2, . . . N \} $ For each $ k $ the topology of the space of states is $ (S^1)^{\times(N-k)} \times [ 0 , 1 ] ^{(\times (k)} $ (i.e. an $ N $ dimensional torus for $ k=0 $ and an $ N $ dimensional cube for $ k = N $ ).

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