Preferred Invariant Symplectic Connections on Compact Coadjoint Orbits

Let be the Haag--Kastler net generated by the (2) chiral current algebra at level 1. We classify the SL(2, )-covariant subsystems by showing that they are all fixed points nets ^H for some subgroup H of the gauge automorphisms group SO(3) of . Then, using the fact that the net ₁ generated by the (1) chiral current can be regarded as a subsystem of , we classify the subsystems of ₁. In this case, there are two distinct proper subsystems: the one generated by the energy-momentum tensor and the gauge invariant subsystem .     

Symmetries and Modular Intersections of Von Neumann Algebras

Let be von Neumann algebras acting on a Hilbert space and let be a common cyclic and separating vector. We say that have the modular intersection property with respect to if(1) -half-sided modular inclusions,(2) (If (1) holds the strong limit exists.) We show that under these conditions the modular groups of and generate a 2-dim. Lie group.This observation is the basis for obtaining group representations of Sl(2, )/Z ₂ generated by modular groups.     

BV Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of(W) Strings

Given a simple, simply laced, complex Lie algebra corresponding to the Lie group G, let be thesubalgebra generated by the positive roots. In this Letter we construct aBV algebra whose underlying graded commutative algebra is given by the cohomology, with respect to , of the algebra of regular functions on G with values in . We conjecture that describes the algebra of allphysical (i.e., BRST invariant) operators of the noncritical string. The conjecture is verified in the two explicitly known cases, ₂ (the Virasoro string) and ₃ (the string).     

Observables and Statistical Maps

This article begins with a review of the framework of fuzzy probability theory. The basic structure is given by the -effect algebra of effects (fuzzy events) and the set of probability measures on a measurable space . An observable is defined, where is the value space of X. It is noted that there exists a one-to-one correspondence between states on and elements of and between observables and -morphisms from to . Various combinations of observables are discussed. These include compositions, products, direct products, and mixtures. Fuzzy stochastic processes are introduced and an application to quantum dynamics is considered. Quantum effects are characterized from among a more general class of effects. An alternative definition of a statistical map is given and it is shown that any statistical map has a unique extension to a statistical operator. Finally, various combinations of statistical maps are discussed and their relationships to the corresponding combinations of observables are derived.     

Generalized Cohomologies and the Physical Subspace of the SU(2) WZNW Model

The zero modes of the monodromy extended SU(2) WZNW model give rise to a gauge theory with a finite-dimensional state space. A generalized BRS operator A such that being the height of the current algebra representation) acts in -dimensional indefinite metric space of quantum group invariant vectors. The generalized cohomologies Ker are 1-dimensional. Their direct sum spans the physical subquotient of .     

Universal R-Matrix for Esoteric Quantum Groups

The universal R-matrix for a class of esoteric (nonstandard) quantum groups q(gl(2N+1)) is constructed as a twisting of the universal R-matrix S of the Drinfeld–Nimbo quantum algebras. The main part of the twisting cocycle is chosen to be the canonical element of an appropriate pair of separated Hopf subalgebras (quantized Borel's (N) q (gl(2N+1))), providing the factorization property of . As a result, the esoteric quantum group generators can be expressed in terms of Drinfeld and Jimbo.     

Integrals of Vertex Operators and uantum Shuffles

Given a braided vector space , we show that iterated integrals of operator-valued functions satisfying a certain exchange relation give rise to representations of the quantum shuffle algebra built on . Using the quantum shuffle construction of the 'upper triangular part' of a quantum shuffle, this provides a simple proof of the result of Bouwknegt, MacCarthy and Pilch saying that integrals of vertex operators acting on certain Fock modules give rise to representations of .     

Multipoint Series of Gromov–Witten Invariants of CP1

We derive explicit formulas for the multipoint series of in degree 0 from the Toda hierarchy, using the recursions of the Toda hierarchy. The Toda equation then yields inductive formulas for the higher degree multipoint series of . We also obtain explicit formulas for the Hodge integrals , in the cases i=0 and 1.     

Poisson Structures for Dispersionless Integrable Systems and Associated W-Algebras

In analogy to the KP theory, the second Poisson structure for the dispersionless KP hierarchy can be defined on the space of commutative pseudodifferential operators . The reduction of the Poisson structure to the symplectic submanifold gives rise to W-algebras. In this Letter, we discuss properties of this Poisson structure, its Miura transformation and reductions. We are particularly interested in the following two cases: (a) L is pure polynomial in p with multiple roots and (b) L has multiple poles at finite distance. The w-algebra corresponding to the case (a) is defined as , where means the multiplicity of roots and to the case (b) is defined by where is the multiplicity of poles. We prove that -algebra is isomorphic via a transformation to U(1) with m= . We also give the explicit free fields representations for these W-algebras.     

On the Algebra of Arrow Diagrams

The purpose of this Letter is to develop further the Gauss diagram approach to finite-type link invariants. In this context we introduce Gauss diagrams counterparts to chord diagrams, 4T relation, weight systems arising from Lie algebras, and an algebra of unitrivalent graphs modulo the STU relation. The counterparts, respectively, are arrow diagrams, 6T relation, weights arising from the solutions of the classical Yang–Baxter equation, and an algebra of acyclic arrow graphs (modulo an oriented version of the STU relation). The algebra encodes, in a graphical form, the main properties of Lie bialgebras, similarly to the well-known relation of the algebra of unitrivalent graphs to Lie algebras. We show that the oriented and relations hold, and that is isomorphic to the algebra of arrow diagrams. As an application, we consider an appropriate link-homotopy version of the algebra . Using this algebra, we construct a Gauss diagram invariants of string links up to link-homotopy, with values both in the algebra and in . We observe that this construction gives the universal Milnor's link-homotopy -invariants.     

A Construction of Representations and Quantum Homogeneous Spaces

A simplified construction of representations is presented for the quantized enveloping algebra q ( ), with being a simple complex Lie algebra belonging to one of the four principal series A\ell, B\ell, C\ell or D\ell. The carrier representation space is the quantized algebra of polynomials in antiholomorphic coordinate functions on the big cell of a coadjoint orbit of K where K is the compact simple Lie group with the Lie algebra – the compact form of .     

Bicross-Product Structure of Affine Quantum Groups

We show that the affine quantum group is isomorphic to a bicross-product central extension of the quantum loop group by a quantum cocycle in R-matrix form.     

Conformally Invariant Quantization at Order Three

Let (M, g) be a pseudo-Riemannian manifold and the space of densities of degree on M. Denote the space of differential operators from to of order k and S _k with = – the corresponding space of symbols. We construct (the unique) conformally invariant quantization map . This result generalizes that of Duval and Ovsienko.     

Chains of twists and induced deformations

Chains of extended twists are composed of factors . The set of Jordanian twists { } can be applied to the initial Hopf algebra . In this case the remaining (transformed) factors of the chain can serve as extensions for such a multijordanian twist. We study the properties of these generalized extensions and the spectra of deformations of the corresponding Heisenberg-like algebras. The results are explicitly demonstrated for the case when .     

On Continuous and Differential Hochschild Cohomology

We show that there are canonical isomorphisms between Hochschild cohomology spaces , where is the algebra of smooth functions on a manifold M and the space of skew multivector fields over M. This implies that continuous and differential deformation theories of coincide.     

Projectively Equivariant Symbol Calculus

The spaces of linear differential operators acting on -densities on and the space of functions on which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect (ⁿ) of vector fields of ⁿ. However, these modules are isomorphic as sl(n + 1,)-modules where is the Lie algebra of infinitesimal projective transformations. In addition, such an -equivariant bijection is unique (up to normalization). This leads to a notion of projectively equivariant quantization and symbol calculus for a manifold endowed with a (flat) projective structure. We apply the -equivariant symbol map to study the of kth-order linear differential operators acting on -densities, for an arbitrary manifold M and classify the quotient-modules .     

Examples of Twisted Cyclic Cocycles from Covariant Differential Calculi

For two covariant differential *-calculi, the twisted cyclic cocycle associated with the volume form is represented in terms of commutators for some self-adjoint operator and some *-representation of the underlying *-algebra.     

\mathfrak{g}-Relative Star Products

We consider Kontsevich star products on the duals of Lie algebras. Such a star product is relative if, for any Lie algebra, its restriction to invariant polynomial functions is the usual pointwise product. Let be a fixed Lie algebra. We shall say that a Kontsevich star product is -relative if, on ^*, its restriction to invariant polynomial functions is the usual pointwise product. We prove that, if is a semi-simple Lie algebra, the only strict Kontsevich -relative star products are the relative (for every Lie algebras) Kontsevich star products.     

Multiplicative Cellular Automata on Nilpotent Groups: Structure, Entropy, and Asymptotics

If , and is a finite (nonabelian) group, then is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of . We characterize when MCA are group endomorphisms of , and show that MCA on inherit a natural structure theory from the structure of . We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.