As was shown in \cite{GPS} the matrix $L=|| l_i^j||$ whose entries $l_i^j$ are generators of the so-called reflection equation algebra is subject to some polynomial identity looking like 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 being a "non-commutative analogue" of the reflection equation algebra. In an appropriate limit we get 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 mentioned above algebras both in the quasiclassical case (i.e. that related to the quantum group $U_q(sl(n))$) and a non-quasiclassical one (i.e. that arising from a Hecke symmetry with non-standard Poincaré series of the corresponding symmetric and skewsymmetric algebras).
Cayley--Hamilton Theorem for Orthogonal Quantum Matrix Algebras
I.E.Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute（I.E.塔姆理论物理系，P.N.列别捷夫物理研究所）；National Research University ”Higher School of Economics”（国立研究型大学高等经济大学）；Bogoliubov Laboratory of Theoretical Physics, JINR（博戈留博夫理论物理实验室，JINR）
