34 research outputs found
On quantum group SL_q(2)
We start with the observation that the quantum group SL_q(2), described in
terms of its algebra of functions has a quantum subgroup, which is just a usual
Cartan group.
Based on this observation we develop a general method of constructing quantum
groups with similar property. We also describe this method in the language of
quantized universal enveloping algebras, which is another common method of
studying quantum groups.
We carry our method in detail for root systems of type SL(2); as a byproduct
we find a new series of quantum groups - metaplectic groups of SL(2)-type.
Representations of these groups can provide interesting examples of bimodule
categories over monoidal category of representations of SL_q(2).Comment: plain TeX, 19 pages, no figure
Duality between quantum symmetric algebras
Using certain pairings of couples, we obtain a large class of two-sided
non-degenerated graded Hopf pairings for quantum symmetric algebras.Comment: 15 pages. Letters in Math. Phy., to appear soo
Braided Hopf algebras obtained from coquasitriangular Hopf algebras
Let be a coquasitriangular Hopf algebra, not necessarily finite
dimensional. Following methods of Doi and Takeuchi, which parallel the
constructions of Radford in the case of finite dimensional quasitriangular Hopf
algebras, we define , a sub-Hopf algebra of , the finite dual of
. Using the generalized quantum double construction and the theory of Hopf
algebras with a projection, we associate to a braided Hopf algebra
structure in the category of Yetter-Drinfeld modules over .
Specializing to , we obtain explicit formulas which endow
with a braided Hopf algebra structure within the category of
left Yetter-Drinfeld modules over .Comment: 43 pages, 1 figur
The Hopf modules category and the Hopf equation
We study the Hopf equation which is equivalent to the pentagonal equation,
from operator algebras. A FRT type theorem is given and new types of quantum
groups are constructed. The key role is played now by the classical Hopf
modules category. As an application, a five dimensional noncommutative
noncocommutative bialgebra is given.Comment: 30 pages, Letax2e, Comm. Algebra in pres
Bicovariant Quantum Algebras and Quantum Lie Algebras
A bicovariant calculus of differential operators on a quantum group is
constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given
by elements of the pure braid group. These operators --- the `reflection
matrix' being a special case --- generate algebras that
linearly close under adjoint actions, i.e. they form generalized Lie algebras.
We establish the connection between the Hopf algebra formulation of the
calculus and a formulation in compact matrix form which is quite powerful for
actual computations and as applications we find the quantum determinant and an
orthogonality relation for in .Comment: 38 page
Generalized diagonal crossed products and smash products for quasi-Hopf algebras. Applications
In this paper we introduce generalizations of diagonal crossed products,
two-sided crossed products and two-sided smash products, for a quasi-Hopf
algebra H. The results we obtain may be applied to H^*-Hopf bimodules and
generalized Yetter-Drinfeld modules. The generality of our situation entails
that the "generating matrix" formalism cannot be used, forcing us to use a
different approach. This pays off because as an application we obtain an easy
conceptual proof of an important but very technical result of Hausser and Nill
concerning iterated two-sided crossed products.Comment: 41 pages, no figure
Strong Connections on Quantum Principal Bundles
A gauge invariant notion of a strong connection is presented and
characterized. It is then used to justify the way in which a global curvature
form is defined. Strong connections are interpreted as those that are induced
from the base space of a quantum bundle. Examples of both strong and non-strong
connections are provided. In particular, such connections are constructed on a
quantum deformation of the fibration . A certain class of strong
-connections on a trivial quantum principal bundle is shown to be
equivalent to the class of connections on a free module that are compatible
with the q-dependent hermitian metric. A particular form of the Yang-Mills
action on a trivial U\sb q(2)-bundle is investigated. It is proved to
coincide with the Yang-Mills action constructed by A.Connes and M.Rieffel.
Furthermore, it is shown that the moduli space of critical points of this
action functional is independent of q.Comment: AMS-LaTeX, 40 pages, major revision including examples of connections
over a quantum real projective spac
Free q-Schrodinger Equation from Homogeneous Spaces of the 2-dim Euclidean Quantum Group
After a preliminary review of the definition and the general properties of
the homogeneous spaces of quantum groups, the quantum hyperboloid qH and the
quantum plane qP are determined as homogeneous spaces of Fq(E(2)). The
canonical action of Eq(2) is used to define a natural q-analog of the free
Schro"dinger equation, that is studied in the momentum and angular momentum
bases. In the first case the eigenfunctions are factorized in terms of products
of two q-exponentials. In the second case we determine the eigenstates of the
unitary representation, which, in the qP case, are given in terms of Hahn-Exton
functions. Introducing the universal T-matrix for Eq(2) we prove that the
Hahn-Exton as well as Jackson q-Bessel functions are also obtained as matrix
elements of T, thus giving the correct extension to quantum groups of well
known methods in harmonic analysis.Comment: 19 pages, plain tex, revised version with added materia
Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group
Braided groups and braided matrices are novel algebraic structures living in
braided or quasitensor categories. As such they are a generalization of
super-groups and super-matrices to the case of braid statistics. Here we
construct braided group versions of the standard quantum groups . They
have the same FRT generators but a matrix braided-coproduct \und\Delta
L=L\und\tens L where , and are self-dual. As an application, the
degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices
; it is a braided-commutative bialgebra in a braided category. As a
second application, we show that the quantum double D(\usl) (also known as
the `quantum Lorentz group') is the semidirect product as an algebra of two
copies of \usl, and also a semidirect product as a coalgebra if we use braid
statistics. We find various results of this type for the doubles of general
quantum groups and their semi-classical limits as doubles of the Lie algebras
of Poisson Lie groups.Comment: 45 pages. Revised (= much expanded introduction
Twist Deformations of the Supersymmetric Quantum Mechanics
The N-extended Supersymmetric Quantum Mechanics is deformed via an abelian
twist which preserves the super-Hopf algebra structure of its Universal
Enveloping Superalgebra. Two constructions are possible. For even N one can
identify the 1D N-extended superalgebra with the fermionic Heisenberg algebra.
Alternatively, supersymmetry generators can be realized as operators belonging
to the Universal Enveloping Superalgebra of one bosonic and several fermionic
oscillators. The deformed system is described in terms of twisted operators
satisfying twist-deformed (anti)commutators. The main differences between an
abelian twist defined in terms of fermionic operators and an abelian twist
defined in terms of bosonic operators are discussed.Comment: 18 pages; two references adde