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1.
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
2.
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
3.
4.
In this paper we construct noncommutative resolutions of a certain class of Calabi-Yau threefolds studied by Cachazo et al.
(Geometric transitions and N = 1 quiver theories. , 2001). The threefolds under consideration are fibered over a complex plane with the fibers being deformed Kleinian singularities.
The construction is in terms of a noncommutative algebra introduced by Ginzburg (Calabi-Yau algebras. , 2006) which we call the “N = 1 ADE quiver algebra”. 相似文献
5.
Hans Tilgner 《International Journal of Theoretical Physics》1973,7(1):67-75
The symplectic vector spaceE of theq andp's of classical mechanics allows a basis free definition of the Poisson bracket in the symmetric algebra overE. Thus the symmetric algebra overE becomes a Lie algebra, which can be compared with the quantum mechanical Weyl algebra with its commutator Lie structure.
The universality of the Weyl algebra is used to study the well-known ‘classical’ Moyal realisation of the Weyl algebra in
the symmetric algebra. Quantisations are defined as linear mappings of the underlying vector spaces of the two algebras. It
is shown that the classical Lie algebra is −2 graded, whereas the quantum Lie algebra is not. This proves that they are not
isomorphic, and hence there is no Dirac quantisation. 相似文献
6.
The algebra of quantum geodesics obtained by quantizing the coordinates of the Teichmller spaces is the quantumso
q(m) algebra by Nelson and Regge.
Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. 相似文献
7.
The general expression for the static permittivity ε(q, 0) of the Coulomb system in the region of small wave vectors was derived based on exact limit relations. The relation obtained
describes the function ε(q, 0) in both “metal” and “dielectric” states of the Coulomb system. On this basis, the concept of the “true” dielectric is
introduced and the definition of the “true” screening length was discussed. Exact relations were derived for the function
ε(q, 0) in the region of small wave vectors q within the random phase approximation at an arbitrary degeneracy. 相似文献
8.
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
9.
We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen “exchangeability”
(i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance
under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables
, we prove that invariance of the joint distribution of the x
i
’s under quantum permutations is equivalent to the fact that the x
i
’s are identically distributed and free with respect to the conditional expectation onto the tail algebra of the x
i
’s.
Research supported by Discovery and LSI grants from NSERC (Canada) and by a Killam Fellowship from the Canada Council for
the Arts. 相似文献
10.
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
11.
Harold Steinacker 《Communications in Mathematical Physics》1998,192(3):687-706
We study irreducible unitary representations of U
q
(SO(2,1)) and U
q
(SO(2,?3)) for q a root of unity, which are finite dimensional. Among others, unitary representations corresponding to all classical one-particle
representations with integral weights are found for , with M being large enough. In the “massless” case with spin bigger than or equal to 1 in 4 dimensions, they are unitarizable only
after factoring out a subspace of “pure gauges” as classically. A truncated associative tensor product describing unitary
many-particle representations is defined for .
Received: 27 November 1996 / Accepted: 28 July 1997 相似文献
12.
C. Pinzari Y. Watatani--RID=""--> K. Yonetani 《Communications in Mathematical Physics》2000,213(2):331-379
To any periodic and full C
*-dynamical system , an invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond
to positive eigenvectors of s. A Perron–Frobenius type theorem asserts the existence of KMS states at inverse temperatures equals the logarithms of the
inner and outer spectral radii of s (extremal KMS states). Examples arising from subshifts in symbolic dynamics, self-similar sets in fractal geometry and noncommutative
metric spaces are discussed.
Certain subshifts are naturally associated to the system, and criteria for the equality of their topological entropy and inverse
temperatures of extremal KMS states are given.
Unital completely positive maps implemented by partitions of unity {x
j
} of grade 1 are considered, resembling the “canonical endomorphism” of the Cuntz algebras. The relationship between the Voiculescu
topological entropy of and the topological entropy of the associated subshift is studied. Examples where the equality holds are discussed among
Matsumoto algebras associated to non finite type subshifts. In the general case is bounded by the sum of the entropy of the subshift and a suitable entropic quantity of the homogeneous subalgebra. Both
summands are necessary.
The measure-theoretic entropy of , in the sense of Connes–Narnhofer–Thirring, is compared to the classical measure-theoretic entropy of the subshift.
A noncommutative analogue of the classical variational principle for the entropy is obtained for the “canonical endomorphism”
of certain Matsumoto algebras. More generally, a necessary condition is discussed. In the case of Cuntz–Krieger algebras an
explicit construction of the state with maximal entropy from the unique KMS state is done.
Received: 1 February 2000 / Accepted: 23 February 2000 相似文献
13.
J. C. Pérez Bueno 《Czechoslovak Journal of Physics》1997,47(12):1275-1282
It is shown that all members in the family of deformed Hopf algebras corresponding to the graded contractions of the inhomogeneous
algebrasiso(p,q),p +q=N, have a bicrossproduct structure.
Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June
1997.
This research has been partially supported by a research grant from the Spanish CICYT. 相似文献
14.
The treatment of supersymmetry is known to cause difficulties in the C*–algebraic framework of relativistic quantum field theory; several no–go theorems indicate that super–derivations and super–KMS
functionals must be quite singular objects in a C*–algebraic setting. In order to clarify the situation, a simple supersymmetric chiral field theory of a free Fermi and Bose
field defined on is analyzed. It is shown that a meaningful C*–version of this model can be based on the tensor product of a CAR–algebra and a novel version of a CCR–algebra, the “resolvent
algebra”. The elements of this resolvent algebra serve as mollifiers for the super–derivation. Within this model, unbounded
(yet locally bounded) graded KMS–functionals are constructed and proven to be supersymmetric. From these KMS–functionals,
Chern characters are obtained by generalizing formulae of Kastler and of Jaffe, Lesniewski and Osterwalder. The characters
are used to define cyclic cocycles in the sense of Connes’ noncommutative geometry which are “locally entire”.
Dedicated to Daniel Kastler on the occasion of his 80th birthday 相似文献
15.
16.
Zhixiang Wu 《International Journal of Theoretical Physics》2011,50(4):1220-1244
In present paper we define a new kind of weak quantized enveloping algebra of Borcherds superalgebras. We denote this algebra
by wUqt(G)wU_{q}^{\tau}(\mathcal{G}). It is a noncommutative and noncocommutative weak graded Hopf algebra under some additional condition. It has a homomorphic
image which is isomorphic to the usual quantum enveloping algebra Uq(G)U_{q}(\mathcal{G}) of G\mathcal{G}. 相似文献
17.
Hopf Modules and Noncommutative Differential Geometry 总被引:1,自引:0,他引:1
We define a new algebra of noncommutative differential forms for any Hopf algebra with an invertible antipode. We prove that there is a one-to-one correspondence between anti-Yetter–Drinfeld modules, which serve as coefficients for the Hopf cyclic (co)homology, and modules which admit a flat connection with respect to our differential calculus. Thus, we show that these coefficient modules can be regarded as “flat bundles” in the sense of Connes’ noncommutative differential geometry. 相似文献
18.
The FRT quantum Euclidean spaces O
q
N
are formulated in terms of Cartesian generators. The quantum analogs of N-dimensional Cayley-Klein spaces are obtained by contractions and analytical continuations. Noncommutative constant-curvature
spaces are introduced as spheres in the quantum Cayley-Klein spaces. For N = 5 part of them is interpreted as the noncommutative analogs of (1+3) space-time models. As a result the quantum (anti)
de Sitter, Minkowski, Newton, Galilei kinematics with the fundamental length and the fundamental time are suggested.
Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. 相似文献
19.
By analogy with the conventional (q=1) case, a squeezed vacuum state for theq-bosonic oscillator is constructed. It can be shown that this obeys quantum noise relations similar to those found in the undeformed state. Using the unitary displacement operator for theq-boson algebra, we show that it is possible to construct aq-squeezed state which is parameterised by elements of a noncommutative algebra. These states satisfy the Robertson-Schrödinger Uncertainty Relation and can be generalised toq-analogues of correlated coherent states.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.One of the authors (RJM) would like to thank the organizers of this colloquium for giving him the opportunity to attend this meeting. He would also like to thank the Carnegie Trust for support in travelling expenses. 相似文献
20.
Jean-Loup Gervais 《Communications in Mathematical Physics》1990,130(2):257-283
On the unit circle, an infinite family of chiral operators is constructed, whose exchange algebra is given by the universalR-matrix of the quantum groupSL(2)
q
. This establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this
quantum group. The method is to relate the monodromy properties of the operator differential equations satisfied by the generalized
vertex operators with the exchange algebra ofSL(2)
q
. The formulae so derived, which generalize an earlier particular case worked out by Babelon, are remarkably compact and may
be entirely written in terms of “q-deformed” factorials and binomial coefficients.
Laboratoire Propre du Centre National de la Recherche Scientifique, associé à l'école Normale Supérieure et à l'Université
de Paris-Sud 相似文献