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1.
We describe discrete symmetries of two-dimensional Yang–Mills theory with gauge group G associated with outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted G-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang–Mills theory but with gauge group given by an extension of G by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang–Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras. 相似文献
2.
We study the structure of abelian extensions of the group L
q
G of q-differentiable loops (in the Sobolev sense), generalizing from the case of the central extension of the smooth loop group.
This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules
are constructed for the Lie algebra. The construction is extended to the current algebra of the supersymmetric Wess-Zumino-Witten
model. An application to the twisted K-theory on G is discussed. 相似文献
3.
We introduce two types of algebras which include respectively the well known reflection equation (RE) and Faddeev-Reshetikhin-Takhtayan
algebras associated with a quasitriangular Hopf algebra H. We show that these two types of algebras are twist-equivalent. It follows that a RE algebra is a module algebra over a twisted
tensor square of H. We present some applications to the equivariant quantization.
Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002. 相似文献
4.
Using the Mackey theory of induced representations all the unitary continuous irreducible representations of the 4-dimensional Lie group G generated by the canonical variables and a positive definite quadratic hamiltonian are found. These are shown to be in a one to one correspondence with the orbits under G in the dual space G to the Lie algebra G of G, and the representations are obtained from the orbits by inducing from one-dimensional representations provided complex subalgebras are admitted. Thus a construction analogous to that of Kirillov and Bernat gives all the representations of this group.The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research OAR through the European Office Aerospace Research, United States Air Force. 相似文献
5.
It is now well-known [1] that the twisted product on the functions defined on a symplectic manifold, play a fundamental role in an invariant approach of quantum mechanics. We prove here a general existence theorem of such twisted products. If a Lie group G acts by symplectomorphisms on a symplectic manifold and if there is a G-invariant symplectic connection, the manifold admits G-invariant Vey twisted products. In particular, if a homogeneous space G/H admits an invariant linear connection, T
*(G/H) admits a G-invariant Vey twisted product. For the connected Lie group G, the group T
*G admits a symplectic structure, a symplectic connection and a Vey twisted product which are bi-invariant under G. 相似文献
6.
We associate a dynamical r-matrix with any such subalgebra L of a finite dimensional self-dual Lie algebra A for which the scalar product of A remains nondegenerate on L and there exists a nonempty open subset Ľ ⊂ L so that the restriction of (ad λ)εEnd( A) to L
is invertible ∨λε Ľ. This r-matrix is also well-defined if L is the grade zero subalgebra of an affine Lie algebra A obtained from a twisted loop algebra based on a finite dimensional self-dual Lie algebra G. Application of evaluation homomorphisms to the twisted loop algebras yields spectral parameter dependent G ⊗ G-valued dynamical r-matrices that are generalizations of Felder’s elliptic r-matrices.
Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June
2001.
This work was supported in part by the Hungarian National Science Fund (OTKA) under T034170. 相似文献
7.
A classification of idempotents of Clifford algebras C
p,q is presented. It is shown that using isomorphisms between Clifford algebras C
p,q and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one-sided ideals in Clifford algebras. Some low-dimensional examples are discussed. 相似文献
8.
We study some properties of invariant states on a C*-algebra A with a group G of automorphisms. Using the concept of G-factorial state, which is a non-commutative generalization of the concept of ergodic measure, in general wider in scope than G-ergodic state, we show that under a certain abelianity condition on ( A, G), which in particular holds for the quasi-local algebras used in statistical mechanics, two different G-ergodic states are disjoint. We also define the concept of G-factorial linear functional, and show that under the same abelianity condition such a functional is proportional to a G-ergodic state. This generalizes an earlier result for complex ergodic measures. 相似文献
9.
A test group is a pair ( G, T) where G is a partially ordered Abelian group and T is a generative antichain in its positive cone. It is shown here that effect algebras and algebraic test groups are coextensive, and a method for calculating the algebraic closure of a test group is developed. Some computational algorithms for studying finite effect algebras are introduced, and the problem of finding quotients of effect algebras is discussed. 相似文献
10.
It is shown that if G is a subgroup of the group of motions of a given space-time then a tetrad can be chosen which is symmetric under the subgroup if and only if the rank of the generators of G is equal to the order of G. As an example such a tetrad is constructed for the Type N twisting empty space-time admitting a two-parameter group of motions. 相似文献
11.
Physical considerations strongly indicate that spacetime at Planck scales is noncommutative. A popular model for such a spacetime
is the Moyal plane. The Poincaré group algebra acts on it with a Drinfel’d-twisted coproduct, however the latter is not appropriate
for more complicated spacetimes such as those containing Friedman-Sorkin (topological) geons. They have rich diffeomorphisms
and mapping class groups, so that the statistics groups for N identical geons is strikingly different from the permutation group S
N
. We generalise the Drinfel’d twist to (essentially all) generic groups including finite and discrete ones, and use it to
deform the commutative spacetime algebras of geons to noncommutative algebras. The latter support twisted actions of diffeomorphisms
of geon spacetimes and their associated twisted statistics. The notion of covariant quantum fields for geons is formulated
and their twisted versions are constructed from their untwisted counterparts. Non-associative spacetime algebras arise naturally in our analysis. Physical consequences, such as the violation of Pauli’s principle, seem
to be one of the outcomes of such nonassociativity. The richness of the statistics groups of identical geons comes from the
nontrivial fundamental groups of their spatial slices. As discussed long ago, extended objects like rings and D-branes also have similar rich fundamental groups. This work is recalled and its relevance to the present quantum geon context
is pointed out. 相似文献
12.
We study non-associative twisted group algebras over
(\mathbb Z2) n{(\mathbb{Z}_2)^n} with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same
way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions
and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. 相似文献
13.
Principal fibre bundles with connections provide geometrical models of gauge theories. Bundles allow for a global formulation of gauge theories: the potentials used in physics are pull-backs, by means of local sections, of the connection form defined on the total space P of the bundle. Given a representation P of the structure (gauge) group G in a vector space V, one defines a (generalized) Higgs field as a map from P to V, equivariant under the action of G in P. If the image of is an orbit W V of G, then a breaks (spontaneously) the symmetry: the isotropy (little) group of w
0 W is the unbroken group H. The principal bundle P is then reduced to a subbundle Q with structure group H. Gravitation corresponds to a linear connection, i.e. to a connection on the bundle of frames. This bundle has more structure than an abstract principal bundle: it is soldered to the base. Soldering results in the occurrence of torsion. The metric tensor is a Higgs field breaking the symmetry from GL (4, R) to the Lorentz group.Invited talk at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.Work on this paper was supported in part by the Polish Research Programme MR. I. 7.This paper is based in part on the research done in 1976–77 when I was Visiting Professor at the State University of New York at Stony Brook. I thank Chen Ning Yang for encouragement, discussions and hospitality at the Institute for Theoretical Physics, SUSB. I have also learned much from conversations with D. Z. Freedman, A. S. Goldhaber, P. van Nieuwenhuizen, J. Smith, P. K. Townsend, W. I. Weisberger, and D. Wilkinson. 相似文献
14.
We derive a formula for the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical
Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular classes of Lie algebras
with a twisted triangular r-matrix. The special case of r-matrices associated to Frobenius Lie algebras is also studied.
相似文献
15.
We describe the twisting construction with the help of 2-cocycles on Hopf–von Neumann and George Kac algebras; we show that
twisted Kac algebras are again Kac algebras. Using this construction, we give a wide class of new quantizations of the Heisenberg
group and describe several series of non-trivial finite- dimensional -Hopf algebras (Kac algebras) of dimensions 4 n and as twisting of finite groups.
Received: Received: 21 March 1997 / Accepted: 2 June 1997 相似文献
17.
The structure and properties of possible q-Minkowski spaces are reviewed and the corresponding non-commutative differential calculi are developed in detail and compared with already existing proposals. This is done by stressing the covariance properties of these algebras with respect to the corresponding q-deformed Lorentz groups as described by appropriate reflection equations. This allow us to give an unified treatment for different q-Minkowski algebras. Some isomorphisms among the space-time and derivative algebras are demonstrated, and their representations are described briefly. Finally, some, physical consequences and open problems are discussed. 相似文献
18.
The property of some finite W algebras to be the commutant of a particular subalgebra of a simple Lie algebra G is used to construct realizations of G. When Gso(4, 2), unitary representations of the conformal and Poincaré algebras are recognized in this approach, which can be compared to the usual induced representation technics. When Gsp(2, R) or sp(4, R), the anyonic parameter can be seen as the eigenvalue of a W generator in such W representations of G.Presented by P. Sorba at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.P. Sorba would like to express his warm thanks to Professor estmír Burdík for the perfect organisation of the conference. 相似文献
19.
Let G be a connected Poisson-Lie group. We discuss aspects of the question of Drinfel'd: can G be quantized? and give some answers. When G is semisimple (a case where the answer is yes), we introduce quantizable Poisson subalgebras of C ∞( G), related to harmonic analysis on G; they are a generalization of F.R.T. models of quantum groups, and provide new examples of quantized Poisson algebras. 相似文献
20.
A number of new results concerning the multicomponent spin systems studied in the first two papers of this series is derived. These results allow for the determination of all such models with number of components M ? 10. There are shown to be 41 such models, each of which is given in terms of its (permissible) symmetry group. The subgroup structure, isomorphisms and the existence of associated algebras of pair energy functions are all discussed in detail. 相似文献
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