DUALITY IN INFINITE DIMENSIONAL FOCK REPRESENTATIONS

An explicit construction is given for the level one irreducible highest weight representations of completed infinite rank affine Lie algebra c∞.

Bäcklund transformations and lattice systems with g-gauge symmetries

The aim of this paper is to describe some connections between spectral theory in infinite dimensional Lie algebras, deformation theory and linearization of nonlinear dynamical systems. We explain how results from isospectral deformations, cohomology groups and algebraic geometry can be used to obtain insight into integrable systems. Another part will be dedicated to the study of infinite continued fractions and isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view. Also, the notion of algebraically completely integrable systems is explained and techniques to solve such systems are presented. Several nonlinear problems in mathematical physics illustrate these results.

Symmetries of nonlinear systems integrable via inverse scattering transform in three dimension are analysed from a different point of view. Actually the classical approach of Lie Backlund transform is not so effective due to the complexity of the computations involved. So we review here the new approach for finding symmetries on the basis of Lax algebra. In the first part we mention in short the earlier techniques for the study of symmetries and in the second part the present method is reviewed in detail following the earlier work of Li Yi Shen et al., with an eye for application to nonlinear systems in three dimension. The approach is highly effective in finding symmetries of non-local, nonlinear in equations in three dimension, which is otherwise impossible to find out. The methodology is illustrated with the help of three examples. (1) the Harry Dym equation in three dimension, (2) the non-local KP equation and (3) dispersive water wave equation in three dimension. Lastly we demonstrate that these symmetries generate an infinite dimensional Lie algebra through a suitably defined commutator.

Ricci Tensors on Some Infinite Dimensional Lie Algebras

The infinite dimensional Lie algebra i>sln e i>Ai>n−1(1) can be realized in several ways as an algebra of differential operators. The aim of this note is to show that the intertwining operators between the realizations of i>sln corresponding to all partitions of i>n can be described very simply by using combinatorial constructions.

In two seminal papers M. Kontsevich introduced graph homology as a tool to compute the homology of three infinite dimensional Lie algebras, associated to the three operads `commutative,' `associative' and `Lie.' We generalize his theorem to all cyclic operads, in the process giving a more careful treatment of the construction than in Kontsevich's original papers. We also give a more explicit treatment of the isomorphisms of graph homologies with the homology of moduli space and Out(F_r) outlined by Kontsevich. In [`Infinitesimal operations on chain complexes of graphs', Mathematische Annalen, 327 (2003) 545-573] we defined a Lie bracket and cobracket on the commutative graph complex, which was extended in [James Conant, `Fusion and fission in graph complexes', Pac. J. 209 (2003), 219-230] to the case of all cyclic operads. These operations form a Lie bi-algebra on a natural subcomplex. We show that in the associative and Lie cases the subcomplex on which the bi-algebra structure exists carries all of the homology, and we explain why the subcomplex in the commutative case does not.

Lie Algebras - Finite and Infinite Dimensional Lie Algebras and Applications in Physics

Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g., the Calogero-Moser space, can be imbedded as coadjoint orbits in the dual of an appropriate infinite dimensional Lie algebra. In particular, there is an infinitesimally transitive action of the Lie algebra in question on the quiver variety. Our construction is based on an extension of Kontsevich's formalism of `non-commutative Symplectic geometry'. We show that this formalism acquires its most adequate and natural formulation in the much more general framework of P-geometry, a `non-commutative geometry' for an algebra over an arbitrary cyclic Koszul operad.

Borovik found a Lie primitive subgroup of E8(ℂ) isomorphic to (Alt5 × Sym6) : 2. In this note, we provide a short proof of existence and his result that the conjugacy class of this subgroup is the only one among those of non-local Lie primitive subgroups of finite dimensional simple complex Lie groups having a socle with more than one simple factor.

The method of graded contractions, based on the preservation of the automorphisms of finite order, is applied to the affine Kac-Moody algebras and their representations, to yield a new class of infinite dimensional Lie algebras and representations. After the introduction of the horizontal and vertical gradings, and the algorithm to find the horizontal toroidal gradings, I discuss some general properties of the graded contractions, and compare them with the In\"on\"u-Wigner contractions. The example of $\hat A_2$ is discussed in detail.

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified. The talk reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren, and the author.

The analogy between the finite-dimensional spin representation Δ of SO(2n) and the infinite-dimensional representation Δ̃ of Sp(2n,R) is made precise. It is then shown that this analogy can be extended so as to provide a precise link between each finite dimensional unitary irreducible representation of SO(2n) and a corresponding infinite-dimensional unitary irreducible representation of Sp(2n,R). The analogy shows itself at the level of the corresponding characters and difference characters, and involves the use of Schur function methods to express both characters and difference characters of SO(2n) and Sp(2n,R) in terms of characters of irreducible representations of their common subgroup U(n). The analogy is extended still further to cover the explicit decomposition of not only tensor products of Δ and Δ̃ with other unitary irreducible representations of SO(2n) and Sp(2n,R), respectively, but also arbitrary tensor powers of Δ and Δ̃.

The systematic study of CR manifolds originated in two pioneering 1932 papers of Élie Cartan. In the first, Cartan classifies all homogeneous CR 3-manifolds, the most well-known case of which is a one-parameter family of left-invariant CR structures on \(\mathrm {SU}_2= S^3\), deforming the standard ‘spherical’ structure. In this paper, mostly expository, we illustrate and clarify Cartan’s results and methods by providing detailed classification results in modern language for four 3-dimensional Lie groups. In particular, we find that \({\mathrm {SL}_2({\mathbb {R}})}\) admits two one-parameter families of left-invariant CR structures, called the elliptic and hyperbolic families, characterized by the incidence of the contact distribution with the null cone of the Killing metric. Low dimensional complex representations of \({\mathrm {SL}_2({\mathbb {R}})}\) provide CR embedding or immersions of these structures. The same methods apply to all other 3-dimensional Lie groups and are illustrated by descriptions of the left-invariant CR structures for \(\mathrm {SU}_2\), the Heisenberg group, and the Euclidean group.

In this paper, discrete analogues of Euler-Poincaré and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L:TG that are G-invariant. These discrete equations provide `reduced' numerical algorithms which manifestly preserve the symplectic structure. The manifold G × G is used as an approximation of TG, and a discrete Langragian :G × G is constructed in such a way that the G-invariance property is preserved. Reduction by G results in a new `variational' principle for the reduced Lagrangian :G, and provides the discrete Euler-Poincaré (DEP) equations. Reconstruction of these equations recovers the discrete Euler-Lagrange equations developed by Marsden et al (Marsden J E, Patrick G and Shkoller S 1998 Commun. Math. Phys. 199 351-395) and Wendlandt and Marsden (Wendlandt J M and Marsden J E 1997 Physica D 106 223-246) which are naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is shown that when G = SO(n), the DEP and DLP algorithms for a particular choice of the discrete Lagrangian are equivalent to the Moser-Veselov scheme for the generalized rigid body.