Quantum line bundles via Cayley-Hamilton identity

Abstract

As shown by Pyatov and Saponov (1995 J. Phys. A: Math. Gen. 28 4415-21) and Gurevich et al (1997 Lett. Math. Phys. 41 255-64), the matrix L = || lij||, whose entries lij are generators of the so-called reflection equation algebra (REA), is subject to some polynomial identity resembling the Cayley-Hamilton identity for a numerical matrix. Here a similar statement is presented for a matrix whose entries are generators of a filtered algebra that is a `non-commutative analogue' of the REA. In an appropriate limit we obtain a similar statement for the matrix formed by the generators of the algebra U(gl(n)). This property is used to introduce the notion of line bundles over quantum orbits in the spirit of the Serre-Swan approach. The quantum orbits in question are presented explicitly as some quotients of one of the algebras mentioned above both in the quasiclassical case (i.e. that related to the quantum group Uq(sl(n))) and a non-quasiclassical one (i.e. that arising from a Hecke symmetry with non-standard Poincare series of the corresponding symmetric and skew-symmetric algebras).