共查询到20条相似文献,搜索用时 15 毫秒
1.
Ping Xu 《Communications in Mathematical Physics》2002,226(3):475-495
In this paper we consider dynamical r-matrices over a nonabelian base. There are two main results. First, corresponding to
a fat reductive decomposition of a Lie algebra ?=?⊕?, we construct geometrically a non-degenerate triangular dynamical r-matrix
using symplectic fibrations. Second, we prove that a triangular dynamical r-matrix naturally corresponds to a Poisson manifold ?⋆×G. A special type of quantization of this Poisson manifold, called compatible star products in this paper, yields a generalized
version of the quantum dynamical Yang–Baxter equation (or Gervais–Neveu–Felder equation). As a result, the quantization problem
of a general dynamical r-matrix is proposed.
Received: 19 May 2001 / Accepted: 19 November 2001 相似文献
2.
In this paper we provide a general condition for the reducibility of the Reshetikhin–Turaev quantum representations of the
mapping class groups. Namely, for any modular tensor category with a special symmetric Frobenius algebra with a non-trivial
genus one partition function, we prove that the quantum representations of all the mapping class groups built from the modular
tensor category are reducible. In particular, for SU(N) we get reducibility for certain levels and ranks. For the quantum SU(2) Reshetikhin–Turaev theory we construct a decomposition for all even levels. We conjecture this decomposition is a complete
decomposition into irreducible representations for high enough levels. 相似文献
3.
We establish some properties of quantum quasi-shuffle algebras. They include the necessary and sufficient condition for the
construction of the quantum quasi-shuffle product, the universal property, and the commutativity condition. As an application,
we use the quantum quasi-shuffle product to construct a linear basis of T(V), for a special kind of Yang–Baxter algebras (V, m, σ). 相似文献
4.
Shahn Majid 《Communications in Mathematical Physics》2002,225(1):131-170
We construct noncommutative “Riemannian manifold” structures on dual quasitriangular Hopf algebras such as ℂ
q
[SU
2] with its standard bicovariant differential calculus, using the quantum frame bundle approach introduced previously. The
metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the n-bein by the Maurer–Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras
ℂ[G] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂ[S
3] is worked out in full detail and a unique torsion free and cotorsion free or “Levi–Civita” connection is obtained with noncommutative
Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as S
3. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal
bundles of tensor product form.
Received: 22 June 2000 / Accepted: 26 August 2001 相似文献
5.
Claudio Garola 《International Journal of Theoretical Physics》2008,47(1):90-103
The word proposition is used in physics with different meanings, which must be distinguished to avoid interpretational problems. We construct
two languages ℒ
*
(x) and ℒ(x) with classical set-theoretical semantics which allow us to illustrate those meanings and to show that the non-Boolean lattice
of propositions of quantum logic (QL) can be obtained by selecting a subset of p-testable propositions within the Boolean lattice of all propositions associated with sentences of ℒ(x). Yet, the aforesaid semantics is incompatible with the standard interpretation of quantum mechanics (QM) because of known
no-go theorems. But if one accepts our criticism of these theorems and the ensuing SR (semantic realism) interpretation of
QM, the incompatibility disappears, and the classical and quantum notions of truth can coexist, since they refer to different
metalinguistic concepts (truth and verifiability according to QM, respectively). Moreover one can construct a quantum language ℒ
TQ
(x) whose Lindenbaum–Tarski algebra is isomorphic to QL, the sentences of which state (testable) properties of individual samples
of physical systems, while standard QL does not bear this interpretation. 相似文献
6.
Andrzej ?uczak 《International Journal of Theoretical Physics》2010,49(12):3176-3184
We construct quantum stochastic integrals for the integrator being a martingale in a von Neumann algebra, and the integrand—a
suitable process with values in the same algebra, as densely defined operators affiliated with the algebra. In the case of
a finite algebra we allow the integrator to be an L
2-martingale in which case the integrals are L
2-martingales too. 相似文献
7.
8.
In this paper, we study the generalized quantum double construction for paired Hopf algebras with particular attention to
the case when the generalized quantum double is a Hopf algebra with projection. Applying our theory to a coquasitriangular
Hopf algebra (H, σ), we see that H has an associated structure of braided Hopf algebra in the category of Yetter-Drinfeld modules over , where H
σ
is a subHopf algebra of H
0, the finite dual of H. Specializing to the quantum group H = SL
q
(N), we find that H
σ
is , so that the duality between these quantum groups is just the evaluation map. Furthermore, we obtain explicit formulas for
the braided Hopf algebra structure of SL
q
(N) in the category of left Yetter-Drinfeld modules over .
The second author held a postdoctoral fellowship at Mount Allison University from 2005 to 2007 and would like to thank Mount
Allison for their warm hospitality. Support for the first author’s research and partial support for the postdoctoral position
of the second author came from an NSERC Discovery Grant. The second author now holds research support from Grant 434/1.10.2007
of CNCSIS. 相似文献
9.
By considering spectral triples on ${S^{2}_{\mu, c}\,\, (c >0 )}${S^{2}_{\mu, c}\,\, (c >0 )} constructed by Chakraborty and Pal (Commun Math Phys 240(3):447–456, 2000), we show that in general the quantum group of
volume and orientation preserving isometries (in the sense of Bhowmick and Goswami in J Funct Anal 257:2530–2572, 2009) for
a spectral triple of compact type may not have a C*-action, and moreover, it can fail to be a matrix quantum group. It is also proved that the category with objects consisting
of those volume and orientation preserving quantum isometries which induce C*-action on the C* algebra underlying the given spectral triple, may not have a universal object. 相似文献
10.
Haisheng Li 《Communications in Mathematical Physics》2010,296(2):475-523
This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine
algebras and Yangians. In this paper, we study notions of (h/2p){\hbar}-adic nonlocal vertex algebra and (h/2p){\hbar}-adic (weak) quantum vertex algebra, slightly generalizing Etingof-Kazhdan’s notion of quantum vertex operator algebra. For
any topologically free
\mathbb C[[(h/2p)]]{{\mathbb C}\lbrack\lbrack{\hbar}\rbrack\rbrack}-module W, we study (h/2p){\hbar}-adically compatible subsets and (h/2p){\hbar}-adically S{\mathcal{S}}-local subsets of (End W)[[x, x
−1]]. We prove that any (h/2p){\hbar}-adically compatible subset generates an (h/2p){\hbar}-adic nonlocal vertex algebra with W as a module and that any (h/2p){\hbar}-adically S{\mathcal{S}}-local subset generates an (h/2p){\hbar}-adic weak quantum vertex algebra with W as a module. A general construction theorem of (h/2p){\hbar}-adic nonlocal vertex algebras and (h/2p){\hbar}-adic quantum vertex algebras is obtained. As an application we associate the centrally extended double Yangian of
\mathfrak s\mathfrak l2{{\mathfrak s}{\mathfrak l}_{2}} to (h/2p){\hbar}-adic quantum vertex algebras. 相似文献
11.
We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of Frenkel and Hernandez (Math Ann, to appear) and Frenkel and Reshetikhin (Commun Math Phys 197(1):1?C32, 1998). We prove this duality for the Kirillov?CReshetikhin modules and their irreducible tensor products. In the course of the proof we introduce and construct ??interpolating (q, t)-characters?? depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual. 相似文献
12.
For a quantum spin chain or 1D fermionic system, we prove that the Drude weight
D verifies the universal Luttinger liquid relation vs2=D/kv_{s}^{2}=D/\kappa, where κ is the susceptibility and v
s
is the Fermi velocity. This result is proved by rigorous Renormalization Group methods and is true for any weakly interacting system, regardless
its integrability. This paper, combined with Benfatto and Mastropietro (in J. Stat. Phys. 138, 1084–1108, 2010), completes the proof of the Luttinger liquid conjecture for such systems. 相似文献
13.
Quantum Spheres and Projective Spaces as Graph Algebras 总被引:5,自引:0,他引:5
The C
*
-algebras of continuous functions on quantum spheres, quantum real projective spaces, and quantum complex projective spaces
are realized as Cuntz-Krieger algebras corresponding to suitable directed graphs. Structural results about these quantum spaces,
especially about their ideals and K-theory, are then derived from the general theory of graph algebras. It is shown that the quantum even and odd dimensional
spheres are produced by repeated application of a quantum double suspension to two points and the circle, respectively.
Received: 31 January 2001 / Accepted: 29 July 2002 Published online: 7 November 2002
RID="*"
ID="*" Supported by grant No. R04–2001–000–00117–0 from the Korea Science & Engineering Foundation.
RID="**"
ID="**" Partially supported by the Research Management Committee of the University of Newcastle. 相似文献
14.
We describe an approach to the noncommutative instantons on the 4-sphere based on quantum group theory. We quantize the Hopf
bundle ?7→?4 making use of the concept of quantum coisotropic subgroups. The analysis of the semiclassical Poisson–Lie structure of U(4) shows that the diagonal SU(2) must be conjugated to be properly quantized. The quantum coisotropic subgroup we obtain is the standard SU
q
(2); it determines a new deformation of the 4-sphere ∑4
q
as the algebra of coinvariants in ?
q
7. We show that the quantum vector bundle associated to the fundamental corepresentation of SU
q
(2) is finitely generated and projective and we compute the explicit projector. We give the unitary representations of ∑4
q
, we define two 0-summable Fredholm modules and we compute the
Chern–Connes pairing between the projector and their characters. It comes out that even the zero class in cyclic homology
is non-trivial.
Received: 3 January 2001 / Accepted: 14 November 2001 相似文献
15.
Anatolij Dvurečenskij 《Foundations of Physics》2013,43(2):210-224
An observable on a quantum structure is any σ-homomorphism of quantum structures from the Borel σ-algebra of the real line into the quantum structure which is in our case a monotone σ-complete effect algebra with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean σ-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there corresponds a spectral measure. 相似文献
16.
We derive an explicit expression for the Haar integral on the quantized algebra of regular functions ℂ
q
[K] on the compact real form K of an arbitrary simply connected complex simple algebraic group G. This is done in terms of the irreducible ✶-representations of the Hopf ✶-algebra ℂ
q
[K]. Quantum analogs of the measures on the symplectic leaves of the standard Poisson structure on K which are (almost) invariant under the dressing action of the dual Poisson algebraic group K
✶ are also obtained. They are related to the notion of quantum traces for representations of Hopf algebras. As an application
we define and compute explicitly quantum analogs of Harish-Chandra c-functions associated to the elements of the Weyl group of G.
Received: 26 January 2001 / Accepted: 31 May 2001 相似文献
17.
Duality between the coloured quantum group and the coloured quantum algebra corresponding to GL(2) is established. The coloured L
± functionals are constructed and the dual algebra is derived explicitly. These functionals are then employed to give a coloured generalisation of the differential calculus on quantum GL(2) within the framework of the R-matrix approach. 相似文献
18.
We propose an algebraic scheme for quantizing the rational Ruijsenaars-Schneider model in the R-matrix formalism. We introduce a special parametrization of the cotangent bundle over . In new variables the standard symplectic structure is described by a classical (Frobenius) r-matrix and by a new dynamical -matrix. Quantizing both of them we find the quantum L-operator algebra and construct its particular representation corresponding to the rational Ruijsenaars-Schneider system.
Using the dual parametrization of the cotangent bundle we also derive the algebra for the L-operator of the hyperbolic Calogero-Moser system.
Received: 24 January 1997 / Accepted: 17 March 1997 相似文献
19.
Fabio Gavarini 《Communications in Mathematical Physics》1997,184(1):95-117
The present work splits in two parts: first, we perform a straightforward generalization of results from [Re], proving that
quantum groups and their unrestricted specializations at roots of 1, in particular the function algebra F[H] of the Poisson group H dual of G, are braided; second, as a main contribution, we prove the convergence of the (specialized) R-matrix action to a birational automorphism of a -fold ramified covering of when is a primitive -th root of 1, and of a 2-fold ramified covering of H, thus giving a geometric content to the notion of braiding for quantum groups at roots of 1.
Received: 23 April 1996/Accepted: 12 August 1996 相似文献
20.
Alexey Sevostyanov 《Communications in Mathematical Physics》1999,204(1):137-146
We propose a method for reduction of quantum systems with arbitrary first-class constraints. An appropriate mathematical setting
for the problem is the homology of associative algebras. For every such algebra A and subalgebra B with augmentation ɛ there exists a cohomological complex which is a generalization of the BRST one. Its cohomology is an
associative graded algebra Hk
*(A,B) which we call the Hecke algebra of the triple (A,B,ɛ). It acts in the cohomology space H
*(B,V) for every left A module V. In particular the zeroth graded component $Hk^{0}(A,B)$ acts in the space of B invariants of $V$ and provides the reduction of the quantum system.
Received: 15 June 1998 / Accepted: 25 January 1999 相似文献