Contragradient algebras and lax operators associated with simple lie algebras

It is shown that the two-dimensional field theories connected with some representations of compact Lie groups admit the Lax pair in terms of contragradient algebras, associated with the corresponding simple Lie algebras.     

Almost periodic functions on connected locally compact groups

Some automatic continuity conditions for almost periodic functions on connected Lie groups and connected locally compact groups are discussed.     

On the soliton spectrum of field theories with character interactions

It is shown that a soliton ansatz in field theories where the interaction is determined by the characters of compact Lie groups can be reduced to the sine-Gordon equation with higher harmonics. The mass spectrum of the basic solitons and the one-loop quantum corrections to it are found for characters of adjoint representations of compact Lie groups with nontrivial center.     

The local geometry of compact homogeneous Lorentz spaces

In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, Zeghib also described the geometric structure of the manifolds. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation and curvatures. In particular, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.     

Monopoles and spectral curves for arbitrary Lie groups

The definition of the spectral curve of a monopole is extended to any connected, compact, simple Lie groupK. It is found there are rankK curves whose degrees are related to the topological weights of the monopole.     

Kirillov’s character formula,the holomorphic Peter–Weyl theorem,and the Blattner–Kostant–Sternberg pairing

By means of the orbit method we show that, for a compact Lie group, the Blattner–Kostant–Sternberg pairing map, with the constants being appropriately fixed, is unitary. Along the way we establish a holomorphic Peter–Weyl theorem for the complexification of a compact Lie group. Among our crucial tools is Kirillov’s character formula. The basic observation is that the Weyl vector is lurking behind the Kirillov character formula, as well as behind the requisite half-form correction on which the Blatter–Kostant–Sternberg-pairing for the compact Lie group relies, and thus produces the appropriate shift which, in turn, controls the unitarity of the BKS-pairing map. Our methods are independent of heat kernel harmonic analysis, which is used by B. C. Hall to obtain a number of these results [B.C. Hall, The Segal–Bargmann Coherent State Transform for compact Lie groups, J. Funct. Anal. 122 (1994) 103–151; B.C. Hall, Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type, Comm. Math. Phys. 226 (2002) 233–268, quant.ph/0012015].     

On non-linear realizations of the groupS U(2)

The non-linear realizations of compact connected Lie groups are considered mainly from the point of view of algebraic topology. In particular, all homogeneous spaces of the groupS U(2) are listed, the construction of a few non-linear realizations ofS U(2) is given and the orbit structure of linear and non-linear realizations are discussed.On leave of absence from Research Institute for Theoretical Physics, Helsinki, Finland.On leave of absence from Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.     

ON THE IRREDUCIBLE REPRESENTATIONS OF THE SIMPLE LIE GROUPS I——THE TENSOR BASIS FOR THE INFINITESIMAL GENERATORS OF THE CLASSICAL SIMPLE LIE GROUPS

In this paper, we analyse the commutation relations of the infinitesimal generatorsof all simple classical Lie groups and establish a new basis for these generators, calledthe tensor basis. In tensor basis, the infinitesimal, generators can be written as somescalar operators, some sets of angular momentum operators and some sets of irreducibletensor operators. The commutation relations, of these operators are very simple andhave many regularities. By means of the method that has been used in the earlier papers, "On the irre-ducible representations of the compact simple Lie groups of rank 2, I,II,III" and thetensor basis, all the irreducible representations of the classical simple Lie groups canbe calculated systematically.     

一般情况下二维非线性_模型的等时Kac-Moody代数

在本文中对一般的紧致单纯群和祸合常数 的任意值证明了带Wess-Zumino 项的二维非线性a 模型的等时流代数构成带中心荷的Kac-Mody 代数. 关键词：     

Convergence of Migdal-Kadanoff iterations: A simple and general proof

A short proof is presented showing convergence of Migdal-Kadanoff iterations for a large class of functions on compact connected Lie groups. From our estimate of the rate of convergence, we deduce lower bounds for string tension and mass gap in hierarchical four-dimensional lattice gauge and twodimensional spin models.     

Fusion Rings of Loop Group Representations

We compute the fusion rings of positive energy representations of the loop groups of the simple, simply connected Lie groups.     

Dirac Induction for Loop Groups

Using a coset version of the cubic Dirac operators for affine Lie algebras, we give an algebraic construction of the Dirac induction homomorphism for loop group representations. With this, we prove a homogeneous generalization of the Weyl–Kac character formula and show compatibility with Dirac induction for compact Lie groups.     

Equivariant Comparison of Quantum Homogeneous Spaces

We consider the quantum homogeneous spaces of the q-deformation of simply connected simple compact Lie groups and their Poisson–Lie quantum subgroups. We prove the deformation invariance in the equivariant KK-theory with respect to the translation action by maximal tori. This extends a result of Neshveyev and Tuset to the equivariant setting. As applications, we prove the ring isomorphism of the K-homology of G _q with respect to the coproduct of C(G _q ), and an analogue of the Borsuk–Ulam theorem for quantum spheres.     

Contractions of Lie Bialgebras and Quantum Algebras

Contractions of Lie bialgebras and Hopf algebras are discussed with examples. Especially, it is shown that the Lie bialgebras associated with the compact simple Lie algebras and the quantum doubles associated with the complex simple Lie algebras can be contracted.     

Oscillator-like unitary representations of non-compact groups with a jordan structure and the non-compact groups of supergravity

We give a general bosonic construction of oscillator-like unitary irreducible representations (UIR) of non-compact groups whose coset spaces with respect to their maximal compact subgroups are Hermitian symmetric. With the exception of E₇₍₇₎, they include all the non-compact invariance groups of extended supergravity theories in four dimensions. These representations have the remarkable property that each UIR is uniquely determined by an irreducible representation of the maximal compact subgroup. We study the connection between our construction, the Hermitian symmetric spaces and the Tits-Koecher construction of the Lie algebras of corresponding groups. We then give the bosonic construction of the Lie algebra ofE ₇₍₇₎ in SU(8), SO(8) and U(7) bases and study its properties. Application of our method toE ₇₍₇₎ leads to reducible unitary representations.Dedicated to Feza Gürsey on the occasion of his 60th birthdayAlexander von Humboldt Fellow, on leave from Physics Dept., Bogaziçi University, Istanbul/Turkey: work supported in part by TBTAK, The National Science and Technology Council of Turkey     

Global Gauge Anomalies in Two-Dimensional Bosonic Sigma Models

We revisit the gauging of rigid symmetries in two-dimensional bosonic sigma models with a Wess-Zumino term in the action. Such a term is related to a background closed 3-form H on the target space. More exactly, the sigma-model Feynman amplitudes of classical fields are associated to a bundle gerbe with connection of curvature H over the target space. Under conditions that were unraveled more than twenty years ago, the classical amplitudes may be coupled to the topologically trivial gauge fields of the symmetry group in a way which assures infinitesimal gauge invariance. We show that the resulting gauged Wess-Zumino amplitudes may, nevertheless, exhibit global gauge anomalies that we fully classify. The general results are illustrated on the example of the WZW and the coset models of conformal field theory. The latter are shown to be inconsistent in the presence of global anomalies. We introduce a notion of equivariant gerbes that allow an anomaly-free coupling of the Wess-Zumino amplitudes to all gauge fields, including the ones in non-trivial principal bundles. Obstructions to the existence of equivariant gerbes and their classification are discussed. The choice of different equivariant structures on the same bundle gerbe gives rise to a new type of discrete-torsion ambiguities in the gauged amplitudes. An explicit construction of gerbes equivariant with respect to the adjoint symmetries over compact simply connected simple Lie groups is given.     

On the Integrability of Geodesic Flows of Submersion Metrics

Letters in Mathematical Physics - Suppose we are given a compact Riemannian manifold (Q,g) with a completely integrable geodesic flow. Let G be a compact connected Lie group acting freely on Q by...     

Optical Models and Symmetries from Finite to Continuous

This contribution to the Proceedings bears the same title as the chapter by this author published in Progress in Optics, and recovers the basic construction starting from the compact algebras so(3) or so(4) for 1- and 2-dimensional finite pixellated-screen optics and their contraction to the Euclidean algebras, in which the geometric and wave models find their realization determined by two symmetry subalgebras, but with questions that may prompt further research. Here we follow and question the path from pixellated-screen optics to three-dimensional geometric optics by contraction between Lie algebras and groups.

     

Unitary Sugawara constructions for affine superalgebras

The untwisted affine basic classical Lie superalgebras are considered. Unitary representations can exist only for certain compact type I superalgebras, and for those type I and II superalgebras yielding spinor gradings of rigid two-dimensional conformal transformations, provided that the central extension satisfies certain constraints. The Sugawara construction for these is presented and the value of the Virasoro central charge listed.