THE QUANTUM SYMMETRIES AND ANTISYMMETRIES OF MATRICES AND THE HIGH-DIMENSIONAL REALIZATIONS OF THE QUANTUM GROUPS GL(LAMBDA-QIJ)(N) AND GL(LAMBDA-QIJ)(N)XGL(LAMBDA-QIJ)(N)

In this letter, the (pij)-symmetry and the (qij)-antisymmetry of matri ces are defined, it is proved that the above two quantum symmetries ar e invariant under a GL(lambda iqij)(N)-involution transformation. By t hese quantum symmetries, a (1.2 N-2(N-1))-dimensional and a (1/2N(2)(N -1))-dimensional hyperspaces are constructed, and on them the quantum groups GL(lambda iqij)(N) and GL(lambda iqij)(N) x GL(lambda iqij) (N) are realized respectively.