3,102 research outputs found
Braided modules and reflection equations
We introduce a representation theory of q-Lie algebras defined earlier in
\cite{DG1}, \cite{DG2}, formulated in terms of braided modules. We also discuss
other ways to define Lie algebra-like objects related to quantum groups, in
particular, those based on the so-called reflection equations. We also
investigate the truncated tensor product of braided modules.Comment: 18 pp., Late
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
Double quantization of \cp type orbits by generalized Verma modules
It is known that symmetric orbits in for any simple Lie algebra
are equiped with a Poisson pencil generated by the
Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to
the "canonical" R-matrix. We realize quantization of this Poisson pencil on
\cp type orbits (i.e. orbits in whose real compact form is ) by means of q-deformed Verma modules.Comment: 21 pages, LaTeX, no figure
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 whose entries
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 . 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
) 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
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