Eigenvalue Asymptotics for the Schrödinger Operator with a δ-Interaction on a Punctured Surface

Given n2, we put r=min . Let be a compact, C ^r -smooth surface in ⁿ which contains the origin. Let further be a family of measurable subsets of such that as . We derive an asymptotic expansion for the discrete spectrum of the Schrödinger operator in L ²( ⁿ ), where is a positive constant, as . An analogous result is given also for geometrically induced bound states due to a interaction supported by an infinite planar curve.     

Bound States in a Locally Deformed Waveguide: The Critical Case

We consider the Dirichlet Laplacian for astrip in with one straight boundary and a width , where $f$ is a smooth function of acompact support with a length 2b. We show that in the criticalcase, , the operator has nobound statesfor small .On the otherhand, a weakly bound state existsprovided . In thatcase, there are positive c ₁,c ₂ suchthat the corresponding eigenvalue satisfies for all sufficiently small.     

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 .     

On the Product of Real Spectral Triples

The product of two real spectral triples and , the first of which is necessarily even, was defined by A.Connes as given by and, in the even-even case, by . Generically it is assumed that the real structure obeys the relations , , , where the -sign table depends on the dimension n modulo 8 of the spectral triple. If both spectral triples obey Connes' >-sign table, it is seen that their product, defined in the straightforward way above, does not necessarily obey this -sign table. In this Letter, we propose an alternative definition of the product real structure such that the -sign table is also satisfied by the product.     

Deformation Quantization of Hermitian Vector Bundles

Motivated by deformation quantization, we consider in this paper *-algebras over rings = (i), where is an ordered ring and I²=–1, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) -valued inner product. For A=C (M), M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that there is a natural bijection between the sets of equivalence classes of local Hermitian deformations of C (M) and ( (E)) and that the corresponding deformed algebras are formally Morita equivalent, an algebraic generalization of strong Morita equivalence of C ^*-algebras. We also discuss the semi-classical geometry arising from these deformations.     

Linearity of the Transverse Poisson Structure to a Coadjoint Orbit

We prove a simple formula for the transverse Poisson structure to a coadjoint orbit (in the dual of a Lie algebra ) and use it in examples such as and . We also give a sufficient condition on the isotropy subalgebra of so that the transverse Poisson structureto the coadjoint orbit of is linear.     

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.     

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.     

\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.     

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.     

Classical Dynamical r-Matrices,Poisson Homogeneous Spaces,and Lagrangian Subalgebras

Lu has shown that any dynamical r-matrix for the pair ( , ) naturally induces a Poisson homogeneous structure on G/U. She also proved that if is complex simple, is its Cartan subalgebra and r is quasitriangular, then this correspondence is in fact one-to-one. In this Letter we find some general conditions under which the Lu correspondence is one-to-one. Then we apply this result to describe all triangular Poisson homogeneous structures on G/U for a simple complex group G and its reductive subgroup U containing a Cartan subgroup.     

Vertex Operators for Boundary Algebras

We construct embeddings of boundary algebras into ZF algebras . Since it is known that these algebras are the relevant ones for the study of quantum integrable systems (with boundaries for and without for ), this connection allows to make the link between different approaches of the systems with boundaries. The construction uses the well-bred vertex operators built recently, and is classified by reflection matrices. It relies only on the existence of an R-matrix obeying a unitarity condition, and as such can be applied to any infinite dimensional quantum group.     

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.     

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.     

Commutator Representations of Covariant Differential Calculi on Quantum Groups

Let (, d) be a first-order differential *-calculus on a *-algebra . We say that a pair (, F) of a *-representation of on a dense domain of a Hilbert space and a symmetric operator F on gives a commutator representation of if there exists a linear mapping : L( ) such that (adb) = (a)i[F, (b) ], a, b . Among others, it is shown that each left-covariant *-calculus of a compact quantum group Hopf *-algebra has a faithful commutator representation. For a class of bicovariant *-calculi on , there is a commutator representation such that F is the image of a central element of the quantum tangent space. If is the Hopf *-algebra of the compact form of one of the quantum groups SL _q (n+1), O _q (n), Sp _q (2n) with real trancendental q, then this commutator representation is faithful.     

On the Leibniz Homology of Poisson Algebras

We study the Leibniz homology of the Poisson algebra of polynomial functions over (²ⁿ ,) where is the standard symplectic structure. We identify it with certain highest-weight vectors of some _2n ( )-modules and obtain some explicit result in low degree.     

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).     

Conformal Transformations as Observables

C denotes either the conformal group in 3+1 dimensions, PSO(4, 2), or in one chiral dimension, PSL(2, ). Let U be a unitary, strongly continuous representation of C satisfying the spectrum condition and inducing, by its adjoint action, automorphisms of a von Neumann algebra . We construct the unique inner representation of the universal covering group of C implementing these automorphisms. satisfies the spectrum condition and acts trivially on any U-invariant vector. This means in particular: Conformal transformations of a field theory having positive energy are weak limit points of local observables. Some immediate implications for chiral subnets are given. We propose the name Borchers–Sugawara construction.     

Norm Inequalities for Positive Operators

Let A, B be positive operators on a Hilbert space, z any complex number, m any positive integer, and any unitarily invariant norm. We show that and Some related inequalities are also obtained.