The Paraboson Fock Space and Unitary Irreducible Representations of the Lie Superalgebra {\mathfrak{osp}(1|2n)}

It is known that the defining relations of the orthosymplectic Lie superalgebra are equivalent to the defining (triple) relations of n pairs of paraboson operators . In particular, with the usual star conditions, this implies that the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of . Apart from the simple cases p = 1 or n = 1, these representations had never been constructed due to computational difficulties, despite their importance. In the present paper we give an explicit and elegant construction of these representations V(p), and we present explicit actions or matrix elements of the generators. The orthogonal basis vectors of V(p) are written in terms of Gelfand-Zetlin patterns, where the subalgebra of plays a crucial role. Our results also lead to character formulas for these infinite-dimensional representations. Furthermore, by considering the branching , we find explicit infinite-dimensional unitary irreducible lowest weight representations of and their characters. NIS was supported by a project from the Fund for Scientific Research – Flanders (Belgium) and by project P6/02 of the Interuniversity Attraction Poles Programme (Belgian State – Belgian Science Policy). An erratum to this article can be found at     

Dynamical Noncommutativity and Noether Theorem in Twisted $${\phi^{\star 4}}$$ Theory

A -product is defined via a set of commuting vector fields , and used in a theory coupled to the fields. The -product is dynamical, and the vacuum solution , reproduces the usual Moyal product. The action is invariant under rigid translations and Lorentz rotations, and the conserved energy–momentum and angular momentum tensors are explicitly derived.      

Regular Strongly Typical Blocks of $${\mathcal{O}^{\mathfrak {q}}}$$

We use the technique of Harish-Chandra bimodules to prove that regular strongly typical blocks of the category for the queer Lie superalgebra are equivalent to the corresponding blocks of the category for the Lie algebra .     

Local Resolvent Estimates for <Emphasis Type="Italic">N</Emphasis>-body Stark Hamiltonians

For an N-body Stark Hamiltonian , the resolvent estimate holds uniformly in with Re and Im , where , and is a compact interval. This estimate is well known as the limiting absorption principle. In this paper, we report that by introducing the localization in the configuration space, a refined resolvent estimate holds uniformly in with Re and Im . Dedicated to Professor Hideo Tamura on the occasion of his 60th birthday     

Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral <Emphasis Type="Italic">su</Emphasis>(2) WZNW Model

A zero modes’ Fock space is constructed for the extended chiral WZNW model. It gives room to a realization of the fusion ring of representations of the restricted quantum universal enveloping algebra at an even root of unity, and of its infinite dimensional extension by the Lusztig operators We provide a streamlined derivation of the characteristic equation for the Casimir invariant from the defining relations of A central result is the characterization of the Grothendieck ring of both and in Theorem 3.1. The properties of the fusion ring in are related to the braiding properties of correlation functions of primary fields of the conformal current algebra model.      

Semiclassical Almost Isometry

Let (M , ω , J) be a compact and connected polarized Hodge manifold, an isodrastic leaf of half-weighted Bohr–Sommerfeld Lagrangian submanifolds. We study the relation between the Weinstein symplectic structure of and the asymptotics of the the pull-back of the Fubini–Study form under the projectivization of the so-called BPU maps on .     

On the Uniqueness of Weak Weyl Representations of the Canonical Commutation Relation

Let (T, H) be a weak Weyl representation of the canonical commutation relation (CCR) with one degree of freedom. Namely T is a symmetric operator and H is a self-adjoint operator on a complex Hilbert space satisfying the weak Weyl relation: for all (the set of real numbers), e^−itH D(T) ⊂ D(T) (i is the imaginary unit and D(T) denotes the domain of T) and . In the context of quantum theory where H is a Hamiltonian, T is called a strong time operator of H. In this paper we prove the following theorem on uniqueness of weak Weyl representations: Let be separable. Assume that H is bounded below with and , where is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Then ( is the closure of T) is unitarily equivalent to a direct sum of the weak Weyl representation on the Hilbert space , where is the multiplication operator by the variable and with . Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation . This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from Japan Society for the Promotion of Science (JSPS).     

Gauss Decomposition of the Yangian $$Y({\mathfrak{gl}}_{m|n})$$

We describe a Gauss decomposition for the Yangian of the general linear Lie superalgebra. This gives a connection between this Yangian and the Yangian of the classical Lie superalgebra Y(A(m − 1, n − 1)) (for m ≠ n) defined and studied in papers by Stukopin, and suggests natural definitions for the Yangians and Y(A(n, n)). We also show that the coefficients of the quantum Berezinian generate the centre of the Yangian . This was conjectured by Nazarov in 1991.     

On the Instability Intervals of the Mathieu–Hill Operator

Consider the Mathieu–Hill operator

Hidden Symmetries and Integrable Hierarchy of the $${\mathcal{N}}$$ = 4 Supersymmetric Yang-Mills Equations

We describe an infinite-dimensional algebra of hidden symmetries of supersymmetric Yang-Mills (SYM) theory. Our derivation is based on a generalization of the supertwistor correspondence. Using the latter, we construct an infinite sequence of flows on the solution space of the SYM equations. The dependence of the SYM fields on the parameters along the flows can be recovered by solving the equations of the hierarchy. We embed the SYM equations in the infinite system of the hierarchy equations and show that this SYM hierarchy is associated with an infinite set of graded symmetries recursively generated from supertranslations. Presumably, the existence of such nonlocal symmetries underlies the observed integrable structures in quantum SYM theory. On leave from Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia. Address after October 1st, 2006: Theoretical Physics Group, The Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, United Kingdom.     

Integrable Discrete Nets in Grassmannians

We consider discrete nets in Grassmannians , which generalize Q-nets (maps with planar elementary quadrilaterals) and Darboux nets (-valued maps defined on the edges of such that quadruples of points corresponding to elementary squares are all collinear). We give a geometric proof of integrability (multidimensional consistency) of these novel nets, and show that they are analytically described by the noncommutative discrete Darboux system.      

Quantization of Semi-Classical Twists and Noncommutative Geometry

A problem of defining the quantum analogues for semi-classical twists in U()[[t]] is considered. First, we study specialization at q = 1 of singular coboundary twists defined in Uq ())[[t]] for g being a nonexceptional Lie algebra, then we consider specialization of noncoboundary twists when = and obtain q-deformation of the semiclassical twist introduced by Connes and Moscovici in noncommutative geometry. Mathematics Subject Classification: 16W30, 17B37, 81R50     

On The Structure and Representations of the Insertion–Elimination Lie Algebra

We examine the structure of the insertion–elimination Lie algebra on rooted trees introduced in Connes and Kreimer (Ann. Henri Poincar 3(3):411–433, 2002). It possesses a triangular structure , like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a “lowest weight” . We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible.      

$${\mathsf{N}}$$ -Complexes as Functors,Amplitude Cohomology and Fusion Rules

We consider -complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology (called generalized cohomology by M. Dubois-Violette) only vanishes on injective functors providing a well defined functor on the stable category. For left truncated -complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive -complexes is proved to be isomorphic to an Ext functor of an indecomposable -complex inside the abelian functor category. Finally we show that for the monoidal structure of -complexes a Clebsch-Gordan formula holds, in other words the fusion rules for -complexes can be determined. This work has been supported by the projects PICT 08280 (ANPCyT), UBACYTX169, PIP-CONICET 5099 and the German Academic Exchange Service (DAAD). The second author is a research member of CONICET (Argentina) and a Regular Associate of ICTP Associate Scheme.     

Elliptic Equation with Singular Terms on Riemannian Manifold

In this paper we deal with the following equation: on a three-dimensional Riemannian manifold . We assume that the volume of Σ, the norm , and are small enough. Using a rather simple argument we show the existence of solution to the problem. Dedicated to Gosia and Basia.     

Non-relativistic Limit of a Dirac Polaron in Relativistic Quantum Electrodynamics

A quantum system of a Dirac particle interacting with the quantum radiation field is considered in the case where no external potentials exist. Then the total momentum of the system is conserved and the total Hamiltonian is unitarily equivalent to the direct integral of a family of self-adjoint operators acting in the Hilbert space , where is the Hilbert space of the quantum radiation field. The fiber operator is called the Hamiltonian of the Dirac polaron with total momentum . The main result of this paper is concerned with the non-relativistic (scaling) limit of . It is proven that the non-relativistic limit of yields a self-adjoint extension of a Hamiltonian of a polaron with spin 1/2 in non-relativistic quantum electrodynamics.     

Renormalized Traces and Cocycles on the Algebra of S 1-Pseudo-differential Operators

Using renormalized (or weighted) traces of classical pseudo-differential operators and calculus on formal symbols. We exhibit three cocycles on the Lie algebra of classical pseudo-differential operators $Cl(S^1,\mathbb{C}^n)Using renormalized (or weighted) traces of classical pseudo-differential operators and calculus on formal symbols. We exhibit three cocycles on the Lie algebra of classical pseudo-differential operators acting on . We first show that the Schwinger functional associated to the Dirac operator is a cocycle on , and not only on a restricted algebra Then, we investigate two bilinear functionals and , which satisfies We show that and are two cocycles in , and and have the same nonvanishing cohomology class. We finaly calculate on classical pseudo-differential operators of order 1 and on differential operators of order 1, in terms of partial symbols. By this last computation, we recover the Virasoro cocyle and the K?hler form of the loop group. Mathematics Subject Classification (1991). 47G30, 47N50     

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).     

On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization

We study classical twists of Lie bialgebra structures on the polynomial current algebra , where is a simple complex finite-dimensional Lie algebra. We focus on the structures induced by the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. It turns out that quasi-trigonometric r-matrices fall into classes labelled by the vertices of the extended Dynkin diagram of . We give the complete classification of quasi-trigonometric r-matrices belonging to multiplicity free simple roots (which have coefficient 1 in the decomposition of the maximal root). We quantize solutions corresponding to the first root of .