Quantum orbits of R-matrix type

Given a simple Lie algebra \gggg, we consider the orbits in \gggg^* which are of R-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-called R-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of R-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pencil on these orbits in a similar way. The notions of q-deformed Lie brackets, braided coadjoint vector fields and tangent vector fields are discussed as well.Comment: 18 pp., Late

Braided algebras and their applications to Noncommutative Geometry

We introduce the notion of a braided algebra and study some examples of these. In particular, R-symmetric and R-skew-symmetric algebras of a linear space V equipped with a skew-invertible Hecke symmetry R are braided algebras. We prove the "mountain property" for the numerators and denominators of their Poincare-Hilbert series (which are always rational functions). Also, we further develop a differential calculus on modified Reflection Equation algebras. Thus, we exhibit a new form of the Leibniz rule for partial derivatives on such algebras related to involutive symmetries R. In particular, we present this rule for the algebra U(gl(m)). The case of the algebra U(gl(2)) and its compact form U(u(2)) (which can be treated as a deformation of the Minkowski space algebra) is considered in detail. On the algebra U(u(2)) we introduce the notion of the quantum radius, which is a deformation of the usual radius, and compute the action of rotationally invariant operators and in particular of the Laplace operator. This enables us to define analogs of the Laplace-Beltrami operators corresponding to certain Schwarzschild-type metrics and to compute their actions on the algebra U(u(2)) and its central extension. Some "physical" consequences of our considerations are presented.Comment: LaTeX file, 24 page

Quantum line bundles via Cayley-Hamilton identity

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 Poincar\'e series of the corresponding symmetric and skewsymmetric algebras).Comment: LaTex file, 21 pages, minor correction in Aknowledgemen