36 research outputs found
A locally compact quantum group of triangular matrices
We construct a one parameter deformation of the group of 2×2 upper triangular matrices with determinant
1 using the twisting construction. An interesting feature of this new example of a locally compact quantum
group is that the Haar measure is deformed in a non-trivial way. Also, we give a complete description of
the dual C∗-algebra and the dual comultiplication.Побудовано однопараметричну деформацію групи верхніх трикутних матриць розміру 2 × 2 із детермінантом 1 з використанням конструкції скруту. Цікавою рисою цього нового прикладу локально компактної квантової групи є нетривіальна деформація міри Хаара. Наведено також повний опис дуальної C*-алгебри та дуальної комультиплікації
Multiple noncommutative tori and Hopf algebras
We derive the Kac-Paljutkin finite-dimensional Hopf algebras as finite
fibrations of the quantum double torus and generalize the construction for
quantum multiple tori.Comment: 18 pages; AMSLaTeX (major revision, the construction of dual
rewritten using approach of multiplier Hopf algebras, references added
The distribution of geodesic excursions into the neighborhood of a cone singularity on a hyperbolic 2-orbifold
A generic geodesic on a finite area, hyperbolic 2-orbifold exhibits an
infinite sequence of penetrations into a neighborhood of a cone singularity, so
that the sequence of depths of maximal penetration has a limiting distribution.
The distribution function is the same for all such surfaces and is described by
a fairly simple formula.Comment: 20 page
Twisting and Rieffel's deformation of locally compact quantum groups. Deformation of the Haar measure
We develop the twisting construction for locally compact quantum groups. A
new feature, in contrast to the previous work of M. Enock and the second
author, is a non-trivial deformation of the Haar measure. Then we construct
Rieffel's deformation of locally compact quantum groups and show that it is
dual to the twisting. This allows to give new interesting concrete examples of
locally compact quantum groups, in particular, deformations of the classical
group and of the Woronowicz' quantum group
Approximate Homomorphisms of Ternary Semigroups
A mapping between ternary semigroups will be
called a ternary homomorphism if . In this paper,
we prove the generalized Hyers--Ulam--Rassias stability of mappings of
commutative semigroups into Banach spaces. In addition, we establish the
superstability of ternary homomorphisms into Banach algebras endowed with
multiplicative norms.Comment: 10 page
More on quantum groups from the the quantization point of view
Star products on the classical double group of a simple Lie group and on
corresponding symplectic grupoids are given so that the quantum double and the
"quantized tangent bundle" are obtained in the deformation description.
"Complex" quantum groups and bicovariant quantum Lie algebras are discused from
this point of view. Further we discuss the quantization of the Poisson
structure on symmetric algebra leading to the quantized enveloping
algebra as an example of biquantization in the sense of Turaev.
Description of in terms of the generators of the bicovariant
differential calculus on is very convenient for this purpose. Finally
we interpret in the deformation framework some well known properties of compact
quantum groups as simple consequences of corresponding properties of classical
compact Lie groups. An analogue of the classical Kirillov's universal character
formula is given for the unitary irreducible representation in the compact
case.Comment: 18 page
Classical and Quantum Nambu Mechanics
The classical and quantum features of Nambu mechanics are analyzed and
fundamental issues are resolved. The classical theory is reviewed and developed
utilizing varied examples. The quantum theory is discussed in a parallel
presentation, and illustrated with detailed specific cases. Quantization is
carried out with standard Hilbert space methods. With the proper physical
interpretation, obtained by allowing for different time scales on different
invariant sectors of a theory, the resulting non-Abelian approach to quantum
Nambu mechanics is shown to be fully consistent.Comment: 44 pages, 1 figure, 1 table Minor changes to conform to journal
versio