Compact Simple Lie Group Research Articles

CHERN–SIMONS APPROACH TO THREE-MANIFOLD INVARIANTS

A new, formal, noncombinatorial approach to invariants of three-dimensional manifolds of Reshetikhin, Turaev and Witten in the framework of nonperturbative topological quantum Chern–Simons theory, corresponding to an arbitrary compact simple Lie group, is presented. A direct implementation of surgery instructions in the context of quantum field theory is proposed. An explicit form of the specialization of the invariant to the group SU(2) is shown.

A polynomial invariant of integral homology 3-spheres

In 1988 Witten [W] proposed invariants Zk(M) ∈ ℂ (what we call, quantum G invariants) for a 3-manifold M and any integer k associated with a compact simple Lie group G. The invariant Zk(M) is formally expressed by an integral (Feynman path integral) over the (infinite dimensional) quotient space of the all connections in G-bundles on M modulo gauge transformations. If one believes in Feynman path integrals, one can expect the asymptotic formula of Zk(M) for large k predicted by perturbation theory. As in [W], the asymptotic formula (which is a power series in k−1) is given by a sum of contributions from flat connections, since the integral contains an integrand which is wildly oscillatory apart from flat connections for large k. More precise forms of the asymptotic formula are studied in [AS1], [AS2] and [Ko].

On a generalized uncertainty principle, coherent states, and the moment map

It is shown that, under general circumstances, symplectic G-orbits in a hamiltonian manifold acted on (symplectically) by a Lie group G provide critical points for the norm squared of the moment map. This fact yields a “variational” interpretation of the symplectic orbits appearing in the projective space attached to an irreducible representation of a compact simple Lie group (according to work of Kostant and Sternberg and of Giavarini and Onofri), where the previous function is also related to the invariant uncertainty considered by Delbourgo and Perelomov. A notion of generalized canonical conjugate variables (in the Kähler case) is also presented and used in the framework of a Kähler geometric interpretation of the Heisenberg uncertainty relations (building on the analysis given by Cirelli, Manià and Pizzocchero and by Provost and Vallee); it is proved, in particular, that the generalized coherent states of Rawnsley minimize the uncertainty relations for any pair of generalized canonically conjugate variables.

Central extensions of sphere groups and their Lie algebras

This note is a short comment on central extensions of Lie groups of the form Map(X, G) when X=Sn and G is a compact simple Lie group. The cohomology of Map(X, G) is computed and cases where the Lie algebra is extended but the corresponding Lie group extension is trivial are considered.

Ideas and methods in mathematical analysis, stochastics and applications, edited by S. Albeverio, J.E Fenstrad, H. Holden and T. Lindstrom. Volume 1 ISBN 0-521-41929-8 pp. 509 Volume II ISBN 0-521-41930-1 pp. 542 Cambridge U.P. (1992)

Preface Picture of Raphael Hoegh-Krohn Bibliography of Raphael Hoegh-Krohn 1. On the scientific work of Raphael Hoegh-Krohn Part I. Stochastic Analysis 2. Some Euclidean integer-valued random fields with Markov properties 3. A Beurling-Deny type structure theorem for Dirichlet forms on general state spaces 4. Log-concavity of radial Schroedinger wave functions and convergence of planetesimal diffusions 5. Dirichlet forms, diffusion processes and spectral dimensions for nested fractals 6. Large deviations for weak solutions of stochastic differential equations 7. On a class of probabilistic integrodifferential equations 8. Operator integrals and martingale integrals with general parameter 9. Wick multiplication and Ito-Skorohod stochastic differential equations 10. Multiscale analysis in random dynamics 11. A limiting distribution connected with fractional parts of linear forms 12. Rapport sur les representations chaotiques 13. Markov properties of solutions of stochastic partial differential equations in a finite volume 14. On stochastic evolution equations with non-homogeneous boundary conditions 15. Grey noise 16. Existence of invariant measures for diffusion process with infinite dimensional state space Part II. Infinite Dimensional Groups 17. On nonlinear equations associated with Lie algebras of diffeomorphism groups of two-dimensional manifolds 18. Fields of left-invariant standard Brownian motion processes on a smooth bundle of compact-simple Lie groups 19. Unitary highest weight representations of gauge groups Part III. Operator Algebras 20. Some non-commutative orbifolds 21. Ergodic actions of non-abelian compact groups 22. Positive projections onto Jordan algebras and their enveloping von Neumann algebras Part IV. Nonlinear Analysis and Applications 23. How many singularities can there be in an energy minimizing map from the ball to the sphere? 24. Front tracking for petroleum reservoirs 25. Quasi-periodic, finite-gap solutions of the modified Korteweg-de Vries equations 26. A new representation of soliton solutions of the Kadomtsev-Petviashvili equation 27. On scalar conservation laws in one dimension.

Representations of simple lie groups

We prove the equivalence of the following conditions for a compact simple Lie group G: (1) G admits complex representations; (2) the center of G contains elements of order higher than two; (3) G has Casimir invariants of odd order; (4) the expansion of G in real homology spheres H∗;(G;R) = H∗;(S3× S2k2+1×…× S2kl+1; R) contains exponents of order 4n + 1. Besides, either of these conditions implies also that G has outer automorphisms, but not vice versa. We also characterize quaternionic representations by the fact that the center is not trivial but contains only involutions (z2 = e).

Nonlinearizability of the Iwasawa Poisson Lie structure

We show that Lu and Weinstein's Iwasawa Poisson Lie structure on a simple compact Lie group K is not linearizable when K is not SU(2).

Two-dimensional topological gravity and intersection theory on the moduli space of holomorphic bundles

We define a two-dimensional topological Yang-Mills theory for an arbitrary compact simple Lie group. This theory is defined in terms of intersection theory on the moduli space of flat connections on a two-dimensional surface and corresponds physically to a two-dimensional reduction and truncation of four-dimensional topological Yang-Mills theory. Two-dimensional topological Yang-Mills theory defines a topological matter system and may be naturally coupled to two-dimensional topological gravity. This topological Yang-Mills theory is also closely related to Chern-Simons gauge theory in 2 + 1 dimensions. We also discuss a relation between SL (2, R ) Chern-Simons theory and two-dimensional topological gravity.

On the Homology of SU(n) Instantons

In this paper we study the homology of the moduli spaces of instantons associated to principal SU(n) bundles over the four-sphere. This is accomplished by exploiting an loop structure implicit in the disjoint union of all moduli spaces associated to a fixed SU(n) with arbitrary instanton number and relating these spaces to the known homology structure of the four-fold loop space on BSU(n) . Moduli spaces of instantons (self-dual connections with respect to a conformal class of metrics) associated to principal G-bundles Pk(G) = P over S4 have proven to be basic objects in modern geometry. Here G is any simple compact Lie group and k is the integer that classifies the bundle P and is referred to as the instanton number. We denote these moduli spaces by At (G). There is a natural inclusion (0. 1) ik (G): Z, (G) Qk BG induced by forgetting the self-duality condition where we have identified the moduli space of based gauge equivalence classes of all connections on P with the kth component of the four-fold loop space Q?4BG [5]. In a fundamental paper, Atiyah and Jones [5] studied the inclusion (0.1) for G = SU(2) (when S4 has its standard conformally flat metric) and posed several fundamental questions. In a series of papers Taubes (cf. [24-26]) proved several basic existence theorems, stability theorems in terms of k and provided a basis framework to describe how the topology of Ak changes as k increases. Taubes' work is much more general than we describe here in that he not only studied general Lie groups G but also replaced 54 by an arbitrary compact closed Riemannian four-manifold (with arbitrary conformal class of metric). In [7] it was observed that, over the four-sphere but with G = Sp(n), the disjoint union of .k over all k form a homotopy C4 space and that iterated loop space techniques may be profitably used to study H. (xk) for individual k. Certain computational results may be obtained immediately from these Received by the editors October 17, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C55, 58E05; Secondary 53C57, 55P35.

Algebras for the two-sphere and the three-sphere groups of compact simple Lie groups

The infinite-dimensional Lie algebras corresponding to the Lie groups of smooth maps from two and three spheres to compact simple Lie groups are studied. The problem of their central extension is solved. The problem of the existence of semidirect sum algebras containing these and the algebras of the groups of diffeomorphisms on the two and three spheres is treated.

Some examples of the algebra of flows

The algebra of the group of smooth maps from a manifold M to a compact simple Lie group G is studied for two cases. The first is when M is the double coset SO (d,R)/SO(d+1,R)/SO (d,R), the corresponding maps are those from a d sphere to G that are invariant under left translations by elements from SO (d, R). In the second example, M is a two-dimensional torus. The problem of central extension of these algebras is solved. For the first example, no central extension is possible. For the second, the number of independent central extensions is infinite.

Self-maps of classifying spaces of compact simple Lie groups

Self-maps of classifying spaces of compact simple Lie groups

On groups of smooth maps into a simple compact Lie group

On groups of smooth maps into a simple compact Lie group

Path integral on a group manifold and the lattice gauge theory Hamiltonian

For a quantum mechanical system living on the manifold of a compact simple Lie group we present explicit formulae for the quantum corrections, both in the Hamiltonian and, for the most common time discretization, in the path integral. As a special application of this rather general procedure, we compare, for lattice gauge theories, the path integral corresponding to the Kogut-Susskind Hamiltonian and the Wilson action. The latter is shown to correspond to a very special but elegant way of discretizing the time variable.

Representation of compact Lie groups and quantization by deformation

We realise the *-representation program for compact semi simple Lie groups, using a Kodaira's map and Berezin's symbols. We study the Fourier transform it induces. We show that injectivity of Kodaira's map selects the data of geometric quantization. In this case an inverse of Kodaira's map is given by a moment map. This relates our construction to existing ones.

Unitary positive energy representations of the gauge group

We investigate the positive energy representations (also called highest weight representations) of the gauge groupC∞ (Tv,G0),G0 being a compact simple Lie group, and discuss their unitarity, using the technique of Verma modules constructed from generalized loop algebras (a simple generalization of Kac-Moody affine Lie algebras). We show that the unitarity of the representation imposes severa restrictions in it. In particular, we show, as a part of a more general result, that the gauge group does not admit faithful unitary positive energy representations.

EQUAL TIME KAC-M00DY ALGEBRA FOR GENERAL TWO DIMENSIONAL NON-LINEAR _ 0MODEL

It is shown that for general compact simple Lie groups and arbitrary value of coupling constant λ 0 , the equal time current algebra of the two dimensional non-linear σ model with Wess-Zumino term is a Kac-Moody algebra with center charge.

Uniqueness of certain 3-dimensional homologically volume minimizing sbmanifolds in compact simple Lie groups

The purpose of this paper is to prove uniqueness of certain 3-dimensional homologically volume minimizing submanifolds in compact simple Lie groups. Let G be a connected compact simple Lie group whose rank is greater than 1 and Gx be an analytic subgroup of G associated with the highest root of G. The explicit definitionof Gx will be found in Section 2. Itis well known that the homology class[d] represented by d generates the real homology group Ha(G; R) of G. Furnishing G with a bi-invariantRiemannian metric ,we consider a volume minimizing submanifold contained in the real homology class [d]. Using the notion of calibrationintroduced by Harvey-Lawson [1],the second named author has proved the following theorem in hispaper [5].

Spectra of some domains in compact Lie groups and their applications

In this paper, we determine explicitly the spectra of the Dirichlet problems of some domains in simply connected compact simple Lie groups. As their applications, combining results of Hoffman [6] and Mori [10], we can state some stability conditions of these domains for the standard minimal isometric immersions into unit spheres.

Certain minimal or homologically volume minimizing submanifolds in compact symmetric spaces

In thispaper we shall study minimal submanifolds in compact symmetric spaces and homologically volume minimizing submanifolds in compact simple Lie groups and quanternionic Kahler manifolds. The firstsubjectis studied by computing the second fundamental forms of submanifolds. In Section 2 using the structure theorem of the firstconjugate loci of compact symmetric spaces (Takeuchi [5])we compute the second fundamental form of a certainsubmanifold which is open and dense in the firstconjugate locus of a compact symmetric space and prove the minimality of it. Moreover we show that the submanifold has no geodesic point. The second subjectis studied by using the notion calibration introduced by Harvey and Lawson [2]. This notionisused in Sections 3 and 4. The fundamental 2-form of a Kahler manifold is one of important examples of calibrations. It satisfiesWirtinger's inequality, which can be statedas follows. Let M be a Kahler manifold with fundamental 2-form o). Then