Minor Identities For Quasi-Determinants And Quantum Determinants

. We present several identities involving quasi-minors of noncommutative generic matrices. These identities are specialized to quantum matrices, yielding q-analogues of various classical determinantal formulas. 1. Introduction Defining a "good" noncommutative notion of determinant is a very old problem that can be traced back to Cayley (cf [Ca]). There have been several attempts at this problem since the beginning of this century. However it is only very recently that I.M. Gelfand and V.S. Retakh made a major breakthrough by introducing the concept of quasi-determinant which generalizes within a totally noncommutative framework the classical concept of determinant (see [GR1], [GR2]). Their main idea was to abandon the multiplicativity of commutative determinants and to focus on their properties with respect to inversion. It also happens that quasi-determinants are very closely related to the representation aspect of automata theory initiated by M.P. Schutzenberger (cf [BR], [Sc] for ..