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1.
We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group
associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms
a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants
is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine
their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their
classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem
of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical
case with orthogonal quantum group symmetry. 相似文献
2.
There exists a coassociative and cocommutative coproduct in the linear space spanned by the two algebraic products of a classical
Hamilton algebra (the algebraic structure underlying classical mechanics [1]). The transition from classical to quantum Hamilton
algebra (the algebraic structure underlying quantum mechanics) is anħ-deformation which preserves not only the Lie property of the classical Hamilton algebra but also the coassociativity and
cocommutativity of the above coproduct. By explicit construction we obtain the algebraic structures of theq-deformed Hamilton algebras which preserve the said properties of the coproduct. Some algorithms of these structures are obtained
and their implications discussed. The problem of consistency of time evolution with theq-deformed kinematical structure is discussed. A characteristic distinction between the parametersħ andq is brought out to stress the fact thatq cannot be regarded as a fundamental constant. 相似文献
3.
Ángel Ballesteros Francisco J. Herranz Orlando Ragnisco Mariano Santander 《International Journal of Theoretical Physics》2008,47(3):649-663
The role of curvature in relation with Lie algebra contractions of the pseudo-orthogonal algebras so(p,q) is fully described by considering some associated symmetrical homogeneous spaces of constant curvature within a Cayley–Klein
framework. We show that a given Lie algebra contraction can be interpreted geometrically as the zero-curvature limit of some
underlying homogeneous space with constant curvature. In particular, we study in detail the contraction process for the three
classical Riemannian spaces (spherical, Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three relativistic
((anti-)de Sitter and Minkowskian) spacetimes. Next, from a different perspective, we make use of quantum deformations of
Lie algebras in order to construct a family of spaces of non-constant curvature that can be interpreted as deformations of
the above nine spaces. In this framework, the quantum deformation parameter is identified as the parameter that controls the
curvature of such “quantum” spaces. 相似文献
4.
P. Moylan 《Czechoslovak Journal of Physics》1997,47(12):1251-1258
We show that it is possible to express the basis elements of the Lie algebra of the Euclidean group,E(2), as simple irrational functions of certainq deformed expressions involving the generators of the quantum algebraU
q
(so(2, 1)). We consider implications of these results for the representation theory of the Lie algebra ofE(2). We briefly discess analogous results forU
q
(so(2, 2)).
Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June
1997. 相似文献
5.
C. Juszczak 《Czechoslovak Journal of Physics》2000,50(11):1271-1278
Bialgebra structures compatible with Lie algebrae(2) ⊕u(1) and two-dimensional Weyl algebra are classified. Most of them are noncoboundary. The corresponding Poisson brackets are
also calculated.
Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. 相似文献
6.
A. Horzela 《Czechoslovak Journal of Physics》2000,50(11):1245-1250
The Wigner problem,i.e. the search for quantum mechanical commutation relations consistent with the Heisenberg evolution equations of a given form
is studied. In the framework of recently proposed generalization of the Wigner approach the classical analogy is postulated
for the form of the time evolution equations only and the forms of the time evolution and symmetry generators are nota priori assumed. Instead of that the set of basic, physically justified algebraic relations is required to have a Lie algebra structure.
Here the problem is formulated for the system of two particles interacting harmonically and a noncanonical Lie algebra of
fundamental quantum mechanical quantities, alternative to the standard canonical one, is found. The solution of the nonrelativistic
problem suggests what kind of algebras could be investigated as its relativistic analogues.
Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. 相似文献
7.
Bojko Bakalov Nikolay M. Nikolov Karl-Henning Rehren Ivan Todorov 《Communications in Mathematical Physics》2007,271(1):223-246
The superselection sectors of two classes of scalar bilocal quantum fields in D ≥ 4 dimensions are explicitly determined by working out the constraints imposed by unitarity. The resulting classification
in terms of the dual of the respective gauge groups U(N) and O(N) confirms the expectations based on general results obtained in the framework of local nets in algebraic quantum field theory,
but the approach using standard Lie algebra methods rather than abstract duality theory is complementary. The result indicates
that one does not lose interesting models if one postulates the absence of scalar fields of dimension D−2 in models with global conformal invariance. Another remarkable outcome is the observation that, with an appropriate choice
of the Hamiltonian, a Lie algebra embedded into the associative algebra of observables completely fixes the representation
theory. 相似文献
8.
9.
V. S. Varadarajan 《Letters in Mathematical Physics》1995,34(3):319-326
The theme of doing quantum mechanics on all Abelian groups goes back to Schwinger and Weyl. This theme was studied earlier from the point of view of approximating quantum systems in infinite-dimensional spaces by those associated to finite Abelian groups. This Letter links this theme to deformation quantization, and explores the set of noncommutative associative algebra structures on the Schwartz-Weil algebra of any locally compact separable Abelian group. If the group is a vector space of even dimension over a non-Archimedean local fieldK, there exists a family of noncommutative (Moyal) structures parametrized by the local field and containing membersarbitrarily close to the classical one, although the classical algebra is rigid in the sense of deformation theory. The-products are defined by Fourier integral operators. The problem of constructing sucharithmetic Moyal structures on the algebra of Schwartz-Bruhat functions on manifolds that are locally likeK
2n
is raised.In memory of Julian Schwinger 相似文献
10.
V. K. Dobrev 《Communications in Mathematical Physics》1989,124(2):319-336
We consider singular Verma modules overA
1
(1)
, i.e., Verma modules for which the central charge is equal to minus the dual Coxeter number. We calculate the characters of certain factor modules of these Verma modules. In one class of cases we are able to prove that these factor modules are actually the irreducible highest modules for those highest weights. We introduce new Weyl groups which are infinitely generated abelian groups and are proper subgroups or isomorphic between themselves. Using these Weyl groups we can rewrite the character formulae obtained in the paper in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semisimple Lie algebras (respectively Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras) so that the new Weyl groups play the role of the usual Weyl group (respectively affine Weyl group). 相似文献
11.
In this paper we introduce a kind of “noncommutative neighbourhood” of a semiclassical parameter corresponding to the Planck
constant. This construction is defined as a certain filtered and graded algebra with an infinite number of generators indexed
by planar binary leaf-labelled trees. The associated graded algebra (the classical shadow) is interpreted as a “distortion”
of the algebra of classical observables of a physical system. It is proven that there exists a q-analogue of the Weyl quantization, where q is a matrix of formal variables, which induces a nontrivial noncommutative analogue of a Poisson bracket on the classical
shadow. 相似文献
12.
《Journal of Nonlinear Mathematical Physics》2013,20(2):252-269
Abstract Let M be an odd-dimensional Euclidean space endowed with a contact 1-form α. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vector fields that preserve the contact structure defined by a. If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form a defines a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symbols. This generalized Hamiltonian operator on the space of symbols is invariant with respect to the action of the projective contact algebra sp(2n+2) the algebra of vector fields which preserve both the contact structure and the projective structure of the Euclidean space. These two operators lead to a decomposition of the space of symbols, except for some critical density weights, which generalizes a splitting proposed by V. Ovsienko in [18]. 相似文献
13.
Alexander Schenkel 《General Relativity and Gravitation》2011,43(10):2605-2630
We study the quantum field theory (QFT) of a free, real, massless and curvature coupled scalar field on self-similar symmetric
spacetimes, which are deformed by an abelian Drinfel’d twist constructed from a Killing and a homothetic Killing vector field.
In contrast to deformations solely by Killing vector fields, such as the Moyal-Weyl Minkowski spacetime, the equation of motion
and Green’s operators are deformed. We show that there is a *-algebra isomorphism between the QFT on the deformed and the
formal power series extension of the QFT on the undeformed spacetime. We study the convergent implementation of our deformations
for toy-models. For these models it is found that there is a *-isomorphism between the deformed Weyl algebra and a reduced
undeformed Weyl algebra, where certain strongly localized observables are excluded. Thus, our models realize the intuitive
physical picture that noncommutative geometry prevents arbitrary localization in spacetime. 相似文献
14.
15.
We interpretN=2 superconformal field theories (SCFTs) formulated by Kazama and Suzuki via Goddard-Kent-Olive (GKO) construction from a viewpoint of the Lie algebra cohomology theory for the affine Lie algebra. We determine the cohomology group completely in terms of a certain subset of the affine Weyl group. We find that this subset describing the cohomology group can be obtained from its classical counterpart by the action of the Dynkin diagram automorphisms. Some algebra automorphisms of theN=2 superconformal algebra are also formulated. Utilizing the algebra automorphisms, we study the field identification problem for the branching coefficient modules in the GKO-construction. Also the structure of the Poincaré polynomial defined for eachN=2 theory is revealed.Dedicated to Professor Noboru Tanaka on his sixtieth birthday 相似文献
16.
Catherine Meusburger Karl-Henning Rehren 《Communications in Mathematical Physics》2003,237(1-2):69-85
The gauge invariant observables of the closed bosonic string are quantized in four space-time dimensions by constructing
their quantum algebra in a manifestly covariant approach, respecting all symmetries of the classical observables. The quantum
algebra is the kernel of a derivation on the universal enveloping algebra of an infinite-dimensional Lie algebra. The search
for Hilbert space representations of this algebra is separated from its construction, and postponed.
Received: 26 February 2002 / Accepted: 5 September 2002
Published online: 28 March 2003
RID="⋆"
ID="⋆" The article is based on the diploma thesis of the first author (C.M.) [1] under the supervision of K. Pohlmeyer, Universit?t
Freiburg, completing a project by the second author (K.-H.R.) lying dormant since around 1987.
RID="⋆⋆"
ID="⋆⋆" Present address: Dept. of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK. E-mail:C.Meusburger@ma.hw.ac.uk
Communicated by H. Araki, D. Buchholz and K. Fredenhagen 相似文献
17.
Mathias Pillin 《Communications in Mathematical Physics》1996,180(1):23-38
Based on the vanishing of the second Hochschild cohomology group of the Weyl algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum mechanics. For the case of aq-oscillator there exists a deforming map to the classical algebra. It is shown that the differential calculus on quantum planes with involution, i.e., if one works in position-momentum realization, can be mapped on aq-difference calculus on a commutative real space. Although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not possess a proper group of global transformations. 相似文献
18.
Siddhartha Sen 《Pramana》2002,59(2):351-357
A classical phase space with a suitable symplectic structure is constructed together with functions which have Possion brackets
algebraically identical to the Lie algebra structure of the Lie group SU(3). It is shown that in this phase space there are
two spheres which intersect at one point. Such a system has a representation as an algebraic curve of the form X
3+…=0 in C
3. The curve introduced is singular at the origin in the limit when the radii of the spheres go to zero. A direct connection
between the Lie groups SU(3) and a singular curve in C
3 is thus established. The key step needed to do this was to treat the Lie group as a quantum system and determine its phase
space. 相似文献
19.
C. Grunspan 《Communications in Mathematical Physics》2002,226(3):627-662
We construct a discrete quantum version of the Drinfeld–Sokolov correspondence for the sine-Gordon system. The classical
version of this correspondence is a birational Poisson morphism between the phase space of the discrete sine-Gordon system
and a Poisson homogeneous space. Under this correspondence, the commuting higher mKdV vector fields correspond to the action
of an Abelian Lie algebra. We quantize this picture (1) by quantizing this Poisson homogeneous space, together with the action
of the Abelian Lie algebra, (2) by quantizing the sine-Gordon phase space, (3) by computing the quantum analogues of the integrals
of motion generating the mKdV vector fields, and (4) by constructing an algebra morphism taking one commuting family of derivations
to the other one.
Received: 3 July 2001 / Accepted: 14 December 2001 相似文献
20.
Generalizations of the q-Onsager algebra are introduced and studied. In one of the simplest case and q = 1, the algebra reduces to the one proposed by Uglov–Ivanov. In the general case and q ≠ 1, an explicit algebra homomorphism associated with coideal subalgebras of quantum affine Lie algebras (simply and non-simply
laced) is exhibited. Boundary (soliton non-preserving) integrable quantum Toda field theories are then considered in light
of these results. For the first time, all defining relations for the underlying non-Abelian symmetry algebra are explicitly
obtained. As a consequence, based on purely algebraic arguments all integrable (fixed or dynamical) boundary conditions are
classified. 相似文献