Showing papers on "Operator algebra published in 2006"
TL;DR: In this article, the authors construct CFT operators which are dual to local bulk fields in the semiclassical limit and show that at finite $N$ the number of independent commuting operators localized within a bulk volume saturates the holographic bound.
Abstract: The Lorentzian anti-de Sitter/conformal field theory correspondence implies a map between local operators in supergravity and nonlocal operators in the CFT. By explicit computation we construct CFT operators which are dual to local bulk fields in the semiclassical limit. The computation is done for general dimension in global, Poincar\'e and Rindler coordinates. We find that the CFT operators can be taken to have compact support in a region of the complexified boundary whose size is set by the bulk radial position. We show that at finite $N$ the number of independent commuting operators localized within a bulk volume saturates the holographic bound.
611 citations
TL;DR: The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2pth root of unity is shown to be equivalent to the SL( 2, ↦)-representations on the extended characters of the logarithmic (1, p) conformal field theory model in this article.
Abstract: The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2pth root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the ribbon element determines the decomposition of π into a ``pointwise'' product of two commuting SL(2, ℤ)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2, ℤ)-representation on the (1, p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the (1, p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of .
268 citations
Book•
27 Oct 2006TL;DR: Some linear and multiplicative Preserver problems on operator algebra and function algebra are discussed in this article, as well as local automorphisms and local isometries of Operator algebra and Function algebra.
Abstract: Some Linear and Multiplicative Preserver Problems on Operator Algebras and Function Algebras.- Preservers on Quantum Structures.- Local Automorphisms and Local Isometries of Operator Algebras and Function Algebras.
248 citations
TL;DR: In this article, a logarithmic conformal field model that extends the (p, q ) Virasoro minimal models is defined as the kernel of the two minimal-model screening operators.
218 citations
01 Feb 2006
TL;DR: Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools as mentioned in this paper.
Abstract: Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -- the theory of operator algebras, category theory, etc.. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT. This paper is a survey of AQFT, with an orientation towards foundational topics. In addition to covering the basics of the theory, we discuss issues related to nonlocality, the particle concept, the field concept, and inequivalent representations. We also provide a detailed account of the analysis of superselection rules by Doplicher, Haag, and Roberts (DHR); and we give an alternative proof of Doplicher and Robert's reconstruction of fields and gauge group from the category of physical representations of the observable algebra. The latter is based on unpublished ideas due to J. E. Roberts and the abstract duality theorem for symmetric tensor *-categories, a self-contained proof of which is given in the appendix.
206 citations
TL;DR: In this article, integrable lattice models with non-compact quantum group symmetry (the modular double of ) are studied with the help of the separation of variables method, and the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations (Baxter equation and quantum Wronskian equation) which is characterized by suitable analytic and asymptotic properties.
Abstract: We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of ) including an integrable lattice regularization of the sinh-Gordon model and a non-compact version of the XXZ model. Their fundamental R-matrices are constructed in terms of the non-compact quantum dilogarithm. Our choice of the quantum group representations naturally ensures self-adjointness of the Hamiltonian and the higher integrals of motion. These models are studied with the help of the separation of variables method. We show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations (Baxter equation and quantum Wronskian equation) which is characterized by suitable analytic and asymptotic properties. A key technical tool is the so-called -operator, for which we give an explicit construction. Our results allow us to establish some connections to related results and conjectures on the sinh-Gordon theory in continuous spacetime. Our approach also sheds some light on the relations between massive and massless models (in particular, the sinh-Gordon and Liouville theories) from the point of view of their integrable structures.
197 citations
TL;DR: In this article, the duality properties of twisted crossed product algebra are studied in detail, and applied to T-duality in Type II string theory to obtain the Tdual of a general principal torus bundle with general H-flux, which is in general a nonassociative, noncommutative, algebra.
Abstract: In this paper, we initiate the study of C
*-algebras endowed with a twisted action of a locally compact abelian Lie group , and we construct a twisted crossed product , which is in general a nonassociative, noncommutative, algebra. The duality properties of this twisted crossed product algebra are studied in detail, and are applied to T-duality in Type II string theory to obtain the T-dual of a general principal torus bundle with general H-flux, which we will argue to be a bundle of noncommutative, nonassociative tori. Nonassociativity is interpreted in the context of monoidal categories of modules. We also show that this construction of the T-dual includes the other special cases already analysed in a series of papers.
184 citations
TL;DR: In this article, the authors construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible core presentation can be strictly larger than the dimension of the latter.
Abstract: We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter. These examples are obtained using a bijective correspondence between certain ergodic coactions on C*-algebras and unitary fiber functors on the representation category of a compact quantum group. We classify these unitary fiber functors on the universal orthogonal and unitary quantum groups. The associated C*-algebras and von Neumann algebras can be defined by generators and relations, but are not yet well understood.
160 citations
TL;DR: In this article, the Operator Quantum Error Correction formalism was introduced, which is a new scheme for the error correction of quantum operations that incorporates the known techniques, i.e., the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method, as special cases.
Abstract: This paper is an expanded and more detailed version of the work [1] in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques -- i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method -- as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of "unitarily noiseless subsystems".
150 citations
TL;DR: In this paper, the authors proposed a method of constructing a cubic interaction in massless higher spin gauge theory both in flat and in AdS space-times of arbitrary dimensions, based on the use of oscillator formalism and on the Becchi-Rouet-Stora Tyutin (BRST) technique.
Abstract: We propose a method of construction of a cubic interaction in massless higher spin gauge theory both in flat and in AdS space-times of arbitrary dimensions. We consider a triplet formulation of the higher spin gauge theory and generalize the higher spin symmetry algebra of the free model to the corresponding algebra for the case of cubic interaction. The generators of this new algebra carry indexes which label the three higher spin fields involved into the cubic interaction. The method is based on the use of oscillator formalism and on the Becchi-Rouet-Stora-Tyutin (BRST) technique. We derive general conditions on the form of cubic interaction vertex and discuss the ambiguities of the vertex which result from field redefinitions. This method can in principle be applied for constructing the higher spin interaction vertex at any order. Our results are a first step towards the construction of a Lagrangian for interacting higher spin gauge fields that can be holographically studied.
124 citations
TL;DR: In this article, a constructive procedure for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra is proposed, which is motivated by the unfolded formulation of dynamical equations developed in the higher spin gauge theory and provides a starting point for generalization to nonlinear case.
Abstract: A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra 𝔣. Under certain conditions, 𝔣-invariant systems of differential equations are shown to be associated with 𝔣-modules that are integrable with respect to some parabolic subalgebra of 𝔣. The suggested construction is motivated by the unfolded formulation of dynamical equations developed in the higher spin gauge theory and provides a starting point for generalization to the nonlinear case. It is applied to the conformal algebra 𝔬(M, 2) to classify all linear conformally invariant differential equations in the Minkowski space. Numerous examples of conformal equations are discussed from this perspective.
TL;DR: In this paper, it was shown that a scaling of variables leaves the noncommutative algebra invariant, so that only the self-consistent effective parameters of the model are physically relevant.
Abstract: We consider noncommutative quantum mechanics with phase space noncommutativity. In particular, we show that a scaling of variables leaves the noncommutative algebra invariant, so that only the self-consistent effective parameters of the model are physically relevant. We also discuss the recently proposed relation of direct proportionality between the noncommutative parameters, showing that it has a limited applicability.
Journal Article•
TL;DR: Popa as mentioned in this paper showed that the orbit structure of the Bernoulli action of a property (T) group, completely remembers the group and the action, and this information is even essentially contained in the crossed product von Neumann algebra, yielding the first Von Neumann strong rigidity theorem in the literature.
Abstract: We survey Sorin Popa's recent work on Bernoulli actions. The paper was written on the occasion of the Bourbaki seminar. Using very original methods from operator algebras, Sorin Popa has shown that the orbit structure of the Bernoulli action of a property (T) group, completely remembers the group and the action. This information is even essentially contained in the crossed product von Neumann algebra, yielding the first von Neumann strong rigidity theorem in the literature. The same methods allow Popa to obtain II_1 factors with prescribed countable fundamental group.
TL;DR: The Birkhoff decomposition of Connes and Kreimer can be obtained from a certain Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter operator.
Abstract: We describe a unification of several apparently unrelated factorizations arising from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type decomposition that was obtained from the Baker–Campbell–Hausdorff formula in our study of the Hopf algebra approach of Connes and Kreimer to renormalization in perturbative quantum field theory. There we showed that the Birkhoff decomposition of Connes and Kreimer can be obtained from a certain Baker–Campbell–Hausdorff recursion formula in the presence of a Rota–Baxter operator. We will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for Lie algebras that arose in vertex operator algebra and conformal field theory, and the even-odd decomposition of combinatorial Hopf algebra characters as well as the Lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations.
10 Aug 2006
TL;DR: In this article, it was shown that a countably generated Hilbert C*-module over a C *-algebra can be considered a standard frame if and only if the sum of the elements of a subset of V converges in norm for every x is an element of V.
Abstract: Let V be a countably generated Hilbert C*-module over a C*-algebra A. We prove that a sequence {;f(i) : i is an element of I}; subset of V is a standard frame for V if and only if the sum Sigma(i is an element of I) converges in norm for every x is an element of V and if there are constants C, D > 0 such that C parallel to x parallel to(2) = parallel to Sigma(i is an element of I) parallel to <= D parallel to x parallel to(2) for every x is an element of V. We also prove that surjective adjointable operators preserve standard frames. A class of frames for countably generated Hilbert C*-modules over the C*-algebra of all compact operators on some Hilbert space is discussed.
TL;DR: This work addresses the geometry of the space of Hermitian operators on a finite-dimensional Hilbert space and discusses and gives examples of entanglement measures for quantum composite systems.
Abstract: We address several problems concerning the geometry of the space of Hermitian operators on a finite-dimensional Hilbert space, in particular the geometry of the space of density states and canonical group actions on it. For quantum composite systems we discuss and give examples of entanglement measures.
21 Feb 2006
TL;DR: Modal Kleene algebras provide a unifying semantics for various program calculi and enhance efficient cross-theory reasoning in this class, often in a very concise pointfree style.
Abstract: Modal Kleene algebras are Kleene algebras enriched by forward and backward box and diamond operators. We formalise the symmetries of these operators as Galois connections, complementarities and dualities. We study their properties in the associated operator algebras and show that the axioms of relation algebra are theorems at the operator level. Modal Kleene algebras provide a unifying semantics for various program calculi and enhance efficient cross-theory reasoning in this class, often in a very concise pointfree style. This claim is supported by novel algebraic soundness and completeness proofs for Hoare logic and by connecting this formalism with an algebraic decision procedure.
TL;DR: In this paper, it is conjectured that to a linear perturbation of the commutation relations which define the algebra there corresponds a linear permutation of the gravitational field, and this is shown to be true in the case of a perturbations of Minkowski space-time.
Abstract: A gravitational field can be defined in terms of a moving frame, which when made noncommutative yields a preferred basis for a differential calculus. It is conjectured that to a linear perturbation of the commutation relations which define the algebra there corresponds a linear perturbation of the gravitational field. This is shown to be true in the case of a perturbation of Minkowski space-time.
TL;DR: In this article, it was shown that the graded dimensions of the principal subspaces associated to the standard modules of vertex operator algebras satisfy certain classical recursion formulas of Rogers and Selberg.
Abstract: Using the theory of intertwining operators for vertex operator algebras we show that the graded dimensions of the principal subspaces associated to the standard modules for satisfy certain classical recursion formulas of Rogers and Selberg. These recursions were exploited by Andrews in connection with Gordon’s generalization of the Rogers–Ramanujan identities and with Andrews’ related identities. The present work generalizes the authors’ previous work on intertwining operators and the Rogers–Ramanujan recursion.
07 Feb 2006
TL;DR: A soundness and finiteness theorem is proved for this interpretation of the multiplicative and exponential fragment of linear logic (MELL) and it is shown that Girard's original approach to GoI 1 via operator algebras is exactly captured in this categorical framework.
Abstract: We consider the multiplicative and exponential fragment of linear logic (MELL) and give a geometry of interaction (GoI) semantics for it based on unique decomposition categories. We prove a soundness and finiteness theorem for this interpretation. We show that Girard's original approach to GoI 1 via operator algebras is exactly captured in this categorical framework.
TL;DR: In this article, a purely algebraic algorithm is presented for computation of invariants (generalized Casimir operators) of Lie algebras using the Cartan method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra.
Abstract: A new purely algebraic algorithm is presented for computation of invariants (generalized Casimir operators) of Lie algebras. It uses the Cartan method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. The algorithm is applied, in particular, to computation of invariants of real low-dimensional Lie algebras. A number of examples are calculated to illustrate its effectiveness and to make a comparison with the same cases in the literature. Bases of invariants of the real six-dimensional solvable Lie algebras with four-dimensional nilradicals are newly calculated and listed in a table.
TL;DR: In this paper, it is conjectured that to a linear perturbation of the commutation relations which define the algebra there corresponds a linear permutation of the gravitational field, and this is shown to be true in the case of a perturbations of Minkowski space-time.
Abstract: A gravitational field can be defined in terms of a moving frame, which when made noncommutative yields a preferred basis for a differential calculus. It is conjectured that to a linear perturbation of the commutation relations which define the algebra there corresponds a linear perturbation of the gravitational field. This is shown to be true in the case of a perturbation of Minkowski space-time.
01 Jan 2006
TL;DR: In this article, the weak operator topology closed algebra generated by these operators is called higher rank semigroupoid algebra (HRSSA) and the cycle graph algebras are identified as matricial multivariable function algesbras.
Abstract: We begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hubert space and creation operators that are partial isometries acting on the space. We call the weak operator topology closed algebra generated by these operators a 'higher rank semigroupoid algebra'. A number of examples are discussed in detail, including the single vertex case and higher rank cycle graphs. In particular, the cycle graph algebras are identified as matricial multivariable function algebras. We obtain reflexivity for a wide class of graphs and characterize semisimplicity in terms of the underlying graph.
Posted Content•
TL;DR: In this article, a short introduction to quantum permutation groups and Hopf algebras and their basic properties is given, as well as a discussion of quantum automorphism groups of finite graphs.
Abstract: This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on 4 points; remarks and comments.
TL;DR: In this paper, a family of non-commutative 3-sphere deformations was constructed by using a Heegaard splitting of the topological 3sphere as a guiding principle.
Abstract: We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C∗-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C∗-algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts.
01 Jul 2006
TL;DR: In this article, the authors studied the properties of shifted vertex operator algebras, which are vertex algebraic structures derived from a given theory by shifting the conformal vector.
Abstract: We study the properties of shifted vertex operator algebras, which are vertex algebras derived from a given theory by shifting the conformal vector. In this way, we are able to exhibit large numbers of vertex operator algebras which are regular(rational and C_2-cofinite) and yet are pathological in one way or another.
Posted Content•
TL;DR: In this paper, the authors introduce unitary representations of continuous groupoids on continuous fields of Hilbert spaces, and investigate some properties of these objects and discuss some of the standard constructions from representation theory in this particular context.
Abstract: We introduce unitary representations of continuous groupoids on continuous fields of Hilbert spaces. We investigate some properties of these objects and discuss some of the standard constructions from representation theory in this particular context. An important r\^{ole} is played by the regular representation. We conclude by discussing some operator algebra associated to continuous representations of groupoids; in particular, we analyse the relationship of continuous representations of $G$ and continuous representations of the Banach $*$-category $\hat{L}^{1}(G)$.
TL;DR: Within this modeling framework one can express data clustering models, logic programs, ordinary and stochastic differential equations, branching processes, graph grammars, and stochy chemical reaction kinetics, which makes the framework particularly suitable for applications in machine learning and multiscale scientific modeling.
Abstract: We define a class of probabilistic models in terms of an operator algebra of stochastic processes, and a representation for this class in terms of stochastic parameterized grammars. A syntactic specification of a grammar is formally mapped to semantics given in terms of a ring of operators, so that composition of grammars corresponds to operator addition or multiplication. The operators are generators for the time-evolution of stochastic processes. The dynamical evolution occurs in continuous time but is related to a corresponding discrete-time dynamics. An expansion of the exponential of such time-evolution operators can be used to derive a variety of simulation algorithms. Within this modeling framework one can express data clustering models, logic programs, ordinary and stochastic differential equations, branching processes, graph grammars, and stochastic chemical reaction kinetics. The mathematical formulation connects these apparently distant fields to one another and to mathematical methods from quantum field theory and operator algebra. Such broad expressiveness makes the framework particularly suitable for applications in machine learning and multiscale scientific modeling.
TL;DR: In this paper, a quantum field theory in non-commutative space time was constructed by twisting the algebra of quantum operators of the corresponding quantum field theories in commutative spaces, and the twisted Fock space and $S$-matrix consistent with this algebra were constructed.
Abstract: We construct a quantum field theory in noncommutative space time by twisting the algebra of quantum operators (especially, creation and annihilation operators) of the corresponding quantum field theory in commutative space time. The twisted Fock space and $S$-matrix consistent with this algebra have been constructed. The resultant $S$-matrix is consistent with that of Filk [Tomas Filk, Phys. Lett. B 376, 53 (1996).]. We find from this formulation that the spin-statistics relation is not violated in the canonical noncommutative field theories.