Showing papers on "Operator algebra published in 1993"

The Verlinde algebra and the cohomology of the Grassmannian

Abstract: Thearticleisdevoted toaquantumeld theoryexplanation oftherelationship between theVerlindealgebra ofthegroup U(k)atlevel N k and the\quantum " cohom ology oftheGrassm annian ofcom plex k planesin N space.In x2,Iexplain therelation between theVerlindealgebra and thegauged W ZW m odelof G=G;in x3,Idescribe the quantum cohom ology and itsorigin in a quantumeld theory; and in x4,Ipresenta path integralargum entform apping between them .

Mappings which Preserve Idempotents, Local Automorphisms, and Local Derivations

Abstract: It is proved that linear mappings of matrix algebras which preserve idempotents are Jordan homomorphisms. Applying this theorem we get some results concerning local derivations and local automorphisms. As an another application, the complete description of all weakly continuous linear surjective mappings on standard operator algebras which preserve projections is obtained. We also study local ring derivations on commutative semisimple Banach algebras.

Representation Theory of Analytic Holonomy C* Algebras

Abstract: Integral calculus on the space of gauge equivalent connections is developed. Loops, knots, links and graphs feature prominently in this description. The framework is well--suited for quantization of diffeomorphism invariant theories of connections. The general setting is provided by the abelian C* algebra of functions on the quotient space of connections generated by Wilson loops (i.e., by the traces of holonomies of connections around closed loops). The representation theory of this algebra leads to an interesting and powerful ``duality'' between gauge--equivalence classes of connections and certain equivalence classes of closed loops. In particular, regular measures on (a suitable completion of) connections/gauges are in 1--1 correspondence with certain functions of loops and diffeomorphism invariant measures correspond to (generalized) knot and link invariants. By carrying out a non--linear extension of the theory of cylindrical measures on topological vector spaces, a faithful, diffeomorphism invariant measure is introduced. This measure can be used to define the Hilbert space of quantum states in theories of connections. The Wilson--loop functionals then serve as the configuration operators in the quantum theory.

A theory of tensor products for module categories for a vertex operator algebra, I

Abstract: This is the first part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a ``vertex tensor category'' structure on the category of modules for a suitable vertex operator algebra. The notion of vertex tensor category is essentially a ``complex analogue'' of the notion of symmetric tensor category, and in fact a vertex tensor category produces a braided tensor category in a natural way. The theory applies in particular to many familiar ``rational'' vertex operator algebras, including those associated with WZNW models, minimal models and the moonshine module. In this paper (Part I), we introduce the notions of $P(z)$- and $Q(z)$-tensor product, where $P(z)$ and $Q(z)$ are two special elements of the moduli space of spheres with punctures and local coordinates, and we present the fundamental properties and constructions of $Q(z)$-tensor products.

An r-Matrix Approach to Nonstandard Classes of Integrable Equations

Abstract: Three different decompositions of the algebra of pseudo-differential operators and the corresponding r-matrices are considered. Three associated classes of nonlinear integrable equations in 1 +1 and 2 + 1 dimensions are discussed within the framework of generalized Lax equations and Sato's approach. The 2 +1-dimensional hierarchies are associated with the Kadomtsev-Petviashvili (KP) equation, the modified KP equation and a Dym equation, respectively. Reductions of the general hierarchies lead to other known integrable 2 + 1dimensional equations as well as to a variety of integrable equations in 1 +1 dimensions. It is shown, how the multi-Hamiltonian structure of the 1 + 1-dimensional equations can be obtained from the underlying r-matrices. Further, intimate relations between the equations associated with the three different r-matrices are revealed. The three classes are related by Darboux theorems originating from gauge transformations and reciprocal links of the Lax operators. These connections are discussed on a general level, leading to a unified picture on (reciprocal) Backlund and auto-Backlund transformations for large classes of integrable equations covered by the KP, the modified KP, and the Dym hierarchies. §

States and operators in the spacetime algebra

Abstract: The spacetime algebra (STA) is the natural, representation-free language for Dirac's theory of the electron. Conventional Pauli, Dirac, Weyl, and Majorana spinors are replaced by spacetime multivectors, and the quantum σ- and γ-matrices are replaced by two-sided multivector operations. The STA is defined over the reals, and the role of the scalar unit imaginary of quantum mechanics is played by a fixed spacetime bivector. The extension to multiparticle systems involves a separate copy of the STA for each particle, and it is shown that the standard unit imaginary induces correlations between these particle spaces. In the STA, spinors and operators can be manipulated without introducing any matrix representation or coordinate system. Furthermore, the formalism provides simple expressions for the spinor bilinear covariants which dispense with the need for the Fierz identities. A reduction to2+1 dimensions is given, and applications beyond the Dirac theory are discussed.

Bicovariant quantum algebras and quantum Lie algebras

Abstract: A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun $$(\mathfrak{G}_q )$$ toU q g, given by elements of the pure braid group. These operators—the “reflection matrix”Y≡L + SL − being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation forY inSO q (N).

An integrated approach to ladder and shift operators for the Morse oscillator, radial Coulomb and radial oscillator potentials

Abstract: The Morse oscillator, radial Coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods. These problems correspond to different realizations of the so(2,1) algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems. The resultant transition operators act as ladder, or energy changing, operators in the cases of the Coulomb and harmonic oscillator potentials, whereas they act as shift operators, acting at constant energy, in the case of the Morse potential. This is a consequence of the so(2,1) dynamical symmetry, whereby the Morse Hamiltonian is expressible solely in terms of the Casimir operator of the algebra. An alternative algebraic approach, the use of the method of supersymmetric quantum mechanics, or factorization, produces in each case a set of shift operators. Relations between the various ladder and shift operators may be identified by means of the appropriate mappings, and these results can be generalized so as to relate the one dimensional Morse oscillator to the radial Coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions.

Spectral invariance of dense subalgebras of operator algebras

Abstract: We define the notion of strong spectral invariance for a dense Frechet subalgebra A of a Banach algebra B. We show that if A is strongly spectral invariant in a C*-algebra B, and G is a compactly generated polynomial growth Type R Lie group, not necessarily connected, then the smooth crossed product G ⋊ A is spectral invariant in the C*-crossed product G ⋊ B. Examples of such groups are given by finitely generated polynomial growth discrete groups, compact or connected nilpotent Lie groups, the group of Euclidean motions on the plane, the Mautner group, or any closed subgroup of one of these. Our theorem gives the spectral invariance of G ⋊ A if A is the set of C∞-vectors for the action of G on B, or if B = C0 (M), and A is a set of G-differentiable Schwartz functions on M. This gives many examples of spectral invariant dense subalgebras for the C*-algebras associated with dynamical systems. We also obtain relevant results about exact sequences, subalgebras, tensoring by smooth compact operators, and strong spectral invariance in L1 (G, B).

Diagonal harmonious state representations

Abstract: For simple unweighted shift operators a family of complex eigenvalue eigenstates of the shift down operators, called theharmonious states, is constructed. Every density matrix is realized as a weighted sum of projections to the harmonious states; and the weight distributions serve as quasiprobability densities for normal ordered operators.

Crystal base and q-vertex operators

Abstract: Theq-deformed vertex operators of Frenkel and Reshetikhin are studied in the framework of Kashiwara's crystal base theory. It is shown that the vertex operators preserve the crystal structure, and are naturally labeled by the global crystal base. As an application the one point functions are calculated for the associated elliptic RSOS models, following the scheme of Kang et al. developed for the trigonometric vertex models.

Two-Dimensional Tensor Function Representation for All Kinds of Material Symmetry

Abstract: All kinds of physically possible material symmetry in two-dimensional space were investigated in a recent work of Q.-S. Zheng and J. P. Boehler. In this paper, we establish the complete and irreducible representations with respect to every kind of material symmetry for scalar-, vector-, and second-order tensor-valued functions in two-dimensional space of any finite number of vectors and second-order tensors. These representations allow general invariant forms of physical and constitutive laws of anisotropic materials to be developed in plane problems.

Dense m-convex Frechet Subalgebras of Operator Algebra Crossed Products by Lie Groups

Abstract: Let A be a dense Frechet *-subalgebra of a C*-algebra B. (We do not require Frechet algebras to be m-convex.) Let G be a Lie group, not necessarily connected, which acts on both A and B by *-automorphisms, and let σ be a sub-polynomial function from G to the nonnegative real numbers. If σ and the action of G on A satisfy certain simple properties, we define a dense Frechet *-subalgebra G ⋊σ A of the crossed product L1 (G, B). Our algebra consists of differentiable A-valued functions on G, rapidly vanishing in σ. We give conditions on σ and the action of G on A which imply the m-convexity of the dense subalgebra G ⋊σ A. A locally convex algebra is said to be m-convex if there is a family of submultiplicative seminorms for the topology of the algebra. The property of m-convexity is important for a Frechet algebra, and is useful in modern operator theory. If G acts as a transformation group on a locally compact space M, we develop a class of dense subalgebras for the crossed product L1 (G, C0 (M)), where C0 (M) denotes the continuous functions on M vanishing at infinity with the sup norm topology. We define Schwartz functions S (M) on M, which are differentiable with respect to some group action on M, and are rapidly vanishing with respect to some scale on M. We then form a dense Frechet *-subalgebra G ⋊σ S (M) of rapidly vanishing, G-differentiable functions from G to S (M). If the reciprocal of σ is in Lp (G) for some p, we prove that our group algebras Sσ (G) are nuclear Frechet spaces, and that G ⋊σ A is the projective completion .

Moment maps at the quantum level

Abstract: We introduce the notion of moment maps for quantum groups acting on their module algebras. When the module algebras are quantizations of Poisson manifolds, we prove that the construction at the quantum level is a quantization of that at the semi-classical level. We also prove that the corresponding smashed product algebras are quantizations of the semi-direct product Poisson structures.

A one-parameter family of hamiltonian structures for the KP hierarchy and a continuous deformation of the nonlinear WKP algebra

Abstract: The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting W-algebra is a one-parameter deformation of WKP admitting a central extension for generic values of the parameter, reducing naturally to Wn for special values of the parameter, and contracting to the centrally extended W1+∞, W∞ and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic towKP. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of\(\hat W_\infty\) which contracts to a new nonlinear algebra of the W∞-type.

Crossing Symmetry in Elliptic Solutions of the Yang-Baxter Equation and a New L-operator for Belavin's Solution

Abstract: Some algebraic structures in elliptic solutions of the Yang-Baxter equations are investigated. The author proves the crossing symmetry in Belavin's model (1981) as well as in the An-1(1) face model and constructs a new family of L-operators for Belavin's R-matrix as an application.

On spectral properties of q -oscillator operators

Abstract: The property of self-adjointness of the operatorQ =a + +a - in three types ofq-oscillator algebras is considered. Spectral measures and generalized eigenfunctions ofQ are found in the cases when this operator is bounded. Generalized eigenvectors are expressed in terms ofq-Hermite polynomials. If the operatorQ is unbounded, then its closure $$\bar Q$$ is not self-adjoint. However, in this case, $$\bar Q$$ admits self-adjoint extensions. Deficiency subspaces are one-dimensional. These subspaces are explicitly found.

Skeins and handlebodies

Abstract: The Temperley-Lieb algebra is used to find a base for the vector space that is associated to a closed surface by the Topological Quantum Field Theory corresponding to the original Jones polynomial invariant

Loop representations for (2+1) gravity on a torus

Abstract: We study the loop representation of the quantum theory for 2+1-dimensional general relativity on a manifold M=T2*R, where T2 is the torus, and compare it with the connection representation for this system. In particular, we look at the loop transform in the part of the phase space where the holonomies are boosts, and study its kernel. This kernel is dense in the connection representation, and the transform is not continuous with respect to the natural topologies, even in its domain of definition. Nonetheless, loop representations isomorphic to the connection representation corresponding to this part of the phase space can still be constructed if due care is taken. We present this construction, but note that certain ambiguities remain; in particular, functions of loops cannot be uniquely associated with functions of connections.

The logarithm of the derivative operator and higher spin algebras of W ∞ type

Abstract: We use the notion of the logarithm of the derivative operator to describeW ∞ type algebras as central extensions of the algebra of differential operators. We also provide closed formulae for the truncations ofW 1+∞ to higher spin algebras withs≧M, for allM≧2. The results are extended to matrix valued differential operators, introducing a logarithmic generalization of the Maurer-Cartan cocycle.

Multi-colour braid-monoid algebras

Abstract: We define multi-colour generalizations of braid-monoid algebras and present explicit matrix representations which are related to two-dimensional exactly solvable lattice models of statistical mechanics. In particular, we show that the two-colour braid-monoid algebra describes the Yang-Baxter algebra of the critical dilute A-D-E models which were recently introduced by Warnaar, Nienhuis and Seaton(1992) as well as by Roche(1992). These and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure. It is conjectured that the Baxterization of general multi-colour braid-monoid algebras will lead to the construction of further solvable lattice models.

A combinatorial approach to topological quantum field theories and invariants of graphs

Abstract: The combinatorial state sum of Turaev and Viro for a compact 3-manifold in terms of quantum 6j-symbols is generalized by introducing observables in the form of coloured graphs. They satisfy braiding relations and allow for surgeries and a discussion of cobordism theory. Application of these techniques give the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any Riemann surface with arbitrary coloured punctures.

Unique determination of an inner product by adjointness relations in the algebra of quantum observables

Abstract: It is shown that if a representation of a *-algebra on a vector space V is an irreducible *-representation with respect to some inner product on V, then under appropriate technical conditions this property determines the inner product uniquely up to a constant factor. Ashtekar(1991) has suggested using the condition that a given representation of the algebra of quantum observables is a *-representation to fix the inner product on the space of physical states. This idea is of particular interest for the quantization of gravity where an obvious prescription for defining an inner product is lacking. The results of this paper show rigorously that Ashtekar's criterion does suffice to determine the inner product in very general circumstances. Two versions of the result are proved: a simpler one which only applies to representations by bounded operators and a more general one which allows for unbounded operators. Some concrete examples are worked out in order to illustrate the meaning and range of applicability of the general theorems.