86 research outputs found
Hilbert Space Representation of an Algebra of Observables for q-Deformed Relativistic Quantum Mechanics
Using a representation of the q-deformed Lorentz algebra as differential
operators on quantum Minkowski space, we define an algebra of observables for a
q-deformed relativistic quantum mechanics with spin zero. We construct a
Hilbert space representation of this algebra in which the square of the mass is diagonal.Comment: 13 pages, LMU-TPW 94-
Heisenberg double as braided commutative Yetter-Drinfel'd module algebra over Drinfel'd double in multiplier Hopf algebra case
Based on a pairing of two regular multiplier Hopf algebras and ,
Heisenberg double is the smash product with respect to
the left regular action of on . Let be the
Drinfel'd double, then Heisenberg double is a Yetter-Drinfel'd
-module algebra, and it is also braided commutative by the
braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to
some infinite dimensional cases.Comment: 18 pages. arXiv admin note: text overlap with arXiv:math/0404029 by
other author
U-duality (sub-)groups and their topology
We discuss some consequences of the fact that symmetry groups appearing in
compactified (super-)gravity may be non-simply connected. The possibility to
add fermions to a theory results in a simple criterion to decide whether a
3-dimensional coset sigma model can be interpreted as a dimensional reduction
of a higher dimensional theory. Similar criteria exist for higher dimensional
sigma models, though less decisive. Careful examination of the topology of
symmetry groups rules out certain proposals for M-theory symmetries, which are
not ruled out at the level of the algebra's. We conclude with an observation on
the relation between the ``generalized holonomy'' proposal, and the actual
symmetry groups resulting from E_10 and E_11 conjectures.Comment: LaTeX, 8 pages, 2 tables, 1 figure, uses IOP-style files. Contributed
to the proceedings of the RTN-workshop ``The quantum structure of space-time
and the geometrical nature of the fundamental interactions,'', Copenhagen,
Denmark, september 200
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
Characteristic Relations for Quantum Matrices
General algebraic properties of the algebras of vector fields over quantum
linear groups and are studied. These quantum algebras
appears to be quite similar to the classical matrix algebra. In particular,
quantum analogues of the characteristic polynomial and characteristic identity
are obtained for them. The -analogues of the Newton relations connecting two
different generating sets of central elements of these algebras (the
determinant-like and the trace-like ones) are derived. This allows one to
express the -determinant of quantized vector fields in terms of their
-traces.Comment: 11 pages, latex, an important reference [16] added
Diagrammar and metamorphosis of coset symmetries in dimensionally reduced type IIB supergravity
Studying the reduction of type IIB supergravity from ten to three space-time
dimensions we describe the metamorphosis of Dynkin diagram for gravity line
"caterpillar" into a type IIB supergravity "dragonfly" that is triggered by
inclusion of scalars and antisymmetric tensor fields. The final diagram
corresponds to type IIB string theory E8 global symmetry group which is the
subgroup of the conjectured E11 hidden symmetry group. Application of the
results for getting the type IIA/IIB T-duality rules and for searching for type
IIB vacua solutions is considered.Comment: 9 pp, 7 figs, LATEX; to be published in JETP Let
The topology of U-duality (sub-)groups
We discuss the topology of the symmetry groups appearing in compactified
(super-)gravity, and discuss two applications. First, we demonstrate that for 3
dimensional sigma models on a symmetric space G/H with G non-compact and H the
maximal compact subgroup of G, the possibility of oxidation to a higher
dimensional theory can immediately be deduced from the topology of H. Second,
by comparing the actual symmetry groups appearing in maximal supergravities
with the subgroups of SL(32,R) and Spin(32), we argue that these groups cannot
serve as a local symmetry group for M-theory in a formulation of de Wit-Nicolai
type.Comment: 18 pages, LaTeX, 1 figure, 2 table
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
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