Showing papers on "Algebra representation published in 1990"

Process algebra

Abstract: Process algebra is the study of distributed or parallel syst em by algebraic means. Originating in computer science, process algebra has been extended in re ce t years to encompass not just discrete-event systems, but also continuously evolving ph enomena, resulting in so-called hybrid process algebras. A hybrid process algebra can be used for th e specification, simulation, control and verification of embedded systems in combination with the ir environment, and for any dynamic system in general. As the vehicle of our exposition, we use th e hybrid process algebra χ (Chi). The syntax and semantics of χ are discussed, and it is explained how equational reasoning simplifies tool implementations for simulation and verification. A bot tle filling line example is introduced to illustrate system analysis by means of equational reasonin g.

Continuous and Discrete Modules

Abstract: Continuous and discrete modules are, essentially, generalizations of infective and projective modules respectively. Continuous modules provide an appropriate setting for decomposition theory of von Neumann algebras and have important applications to C*-algebras. Discrete modules constitute a dual concept and are related to number theory and algebraic geometry: they possess perfect decomposition properties. The advantage of both types of module is that the Krull-Schmidt theorem can be applied, in part, to them. The authors present here a complete account of the subject and at the same time give a unified picture of the theory. The treatment is essentially self-contained, with background facts being summarized in the first chapter. This book will be useful therefore either to individuals beginning research, or the more experienced worker in algebra and representation theory.

Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra

Abstract: 0.1. An important role in the theory of modular representations is played by certain finite dimensional Hopf algebras u over Fp (the field with p elements, p = prime). Originally, u was defined (Curtis [3]) as the restricted enveloping algebra of a "simple" Lie algebra over Fp For our purposes, it will be more convenient to define u as follows. Let us fix an indecomposable positive definite symmetric Cartan matrix

Blocks of Tame Representation Type and Related Algebras

Abstract: Algebras, quivers, representation type, Auslander-Reiten theory, coverings.- Special biserial algebras and the local semidihedral algebra.- Tame symmetric local algebras.- More on modules, quivers, Auslander-Reiten sequences.- Stable Auslander-Reiten components for tame blocks.- Algebras of dihedral type.- Algebras of guaternion type.- Algebras of semidihedral type.- Centres, blocks, decomposition numbers.- Some applications.

Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics

Abstract: Given the local observables in the vacuum sector fulfilling a few basic principles of local quantum theory, we show that the superselection structure, intrinsically determined a priori, can always be described by a unique compact global gauge group acting on a field algebra generated by field operators which commute or anticommute at spacelike separations. The field algebra and the gauge group are constructed simultaneously from the local observables. There will be sectors obeying parastatistics, an intrinsic notion derived from the observables, if and only if the gauge group is non-Abelian. Topological charges would manifest themselves in field operators associated with spacelike cones but not localizable in bounded regions of Minkowski space. No assumption on the particle spectrum or even on the covariance of the theory is made. However the notion of superselection sector is tailored to theories without massless particles. When translation or Poincare covariance of the vacuum sector is assumed, our construction leads to a covariant field algebra describing all covariant sectors.

Toroidal Lie algebras and vertex representations

Abstract: The paper describes the theory of the toroidal Lie algebra, i.e. the Lie algebra of polynomial maps of a complex torus ℂ××ℂ× into a finite-dimensional simple Lie algebra g. We describe the universal central extension t of this algebra and give an abstract presentation for it in terms of generators and relations involving the extended Cartan matrix of g. Using this presentation and vertex operators we obtain a large class of integrable indecomposable representations of t in the case that g is of type A, D, or E. The submodule structure of these indecomposable modules is described in terms of the ideal structure of a suitable commutative associative algebra.

Affine Kac-Moody algebras and semi-infinite flag manifolds

Abstract: We study representations of affine Kac-Moody algebras from a geometric point of view. It is shown that Wakimoto modules introduced in [18], which are important in conformal field theory, correspond to certain sheaves on a semi-infinite flag manifold with support on its Schhubert cells. This manifold is equipped with a remarkable semi-infinite structure, which is discussed; in particular, the semi-infinite homology of this manifold is computed. The Cousin-Grothendieck resolution of an invertible sheaf on a semi-infinite flag manifold gives a two-sided resolution of an irreducible representation of an affine algebras, consisting of Wakimoto modules. This is just the BRST complex. As a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent Lie algebra.

Fock representations and BRST cohomology in SL (2) current algebra

Abstract: We investigate the structure of the Fock modules overA1(1) introduced by Wakimoto. We show that irreducible highest weight modules arise as degree zero cohomology groups in a BRST-like complex of Fock modules. Chiral primary fields are constructed as BRST invariant operators acting on Fock modules. As a result, we obtain a free field representation of correlation functions of theSU(2) WZW model on the plane and on the torus. We also consider representations of fractional level arising in Polyakov's 2D quantum gravity. Finally, we give a geometrical, Borel-Weil-like interpretation of the Wakimoto construction.

Area-preserving diffeomorphisms and higher-spin algebras

Abstract: We show that there exists a one-parameter family of infinite-dimensional algebras that includes the bosonicd=3 Fradkin-Vasiliev higher-spin algebra and the non-Euclidean version of the algebra of area-preserving diffeomorphisms of the two-sphereS2 as two distinct members. The non-Euclidean version of the area preserving algebra corresponds to the algebra of area-preserving diffeomorphisms of the hyperbolic spaceS1,1, and can be rewritten as\(\mathop {\lim }\limits_{N \to \infty } su(N,N)\). As an application of our results, we formulate a newd=2+1 massless higher-spin field theory as the gauge theory of the area-preserving diffeomorphisms ofS1,1.

Infinite Dimensional Algebras and a Trigonometric Basis for the Classical Lie Algebras

Abstract: This paper explores features of the infinite‐dimensional algebras that have been previously introduced. In particular, it is shown that the classical simple Lie algebras (AN, BN, CN, DN) may be expressed in an ‘‘egalitarian’’ basis with trigonometric structure constants. The transformation to the standard Cartan–Weyl basis, and the particularly transparent N→∞ limit that this formulation allows is provided.

On Kleene algebras and closed semirings

Abstract: Kleene algebras are an important class of algebraic structures that arise in diverse areas of computer science: program logic and semantics, relational algebra, automata theory, and the design and analysis of algorithms. The literature contains at several inequivalent definitions of Kleene algebras and related algebraic structures [2,13,14,5,6,1,9,7].

Module algebra

Abstract: An axiomatic algebraic calculus of modules is given that is based on the operators combination/union, export, renaming, and taking the visible signature. Four different models of module algebra are discussed and compared.

A Class of Algebras Similar to the Enveloping Algebra of sl(2)

Abstract: Fix f E C[X]. Define R = C[A, B, H] subject to the relations HA-AH = A, HB-BH =-B, AB-BA = f(H). We study these algebras (for different f) and in particular show how they are similar to (and different from) U(sl(2)), the enveloping algebra of sl(2, C). There is a notion of highest weight modules and a category a' for such R. For each n > 0, if f (x) = (x + 1)n+l _ Xn+1 , then R has precisely n simple modules in each finite dimension, and every finite-dimensional R-module is semisimple.

The quantum group structure associated with non-linearly extended Virasoro algebras

Abstract: Recently, an infinite family of chiral Virasoro vertex operators, whose exchange algebra is given by the universalR-matrix ofSL(2) q , has been constructed. In the present paper, the case of non-linearly (W-) extended Virasoro symmetries, related to the algebrasAN,N>1, is considered along the same line. Contrary to the previous case (A1) the standardR-matrix ofSL(N+1)q does not come out, and a different solution of Yang and Baxter's equations is derived. The associated quantum group structure is displayed.

The quantum group structure of 2D gravity and minimal models I

Abstract: On the unit circle, an infinite family of chiral operators is constructed, whose exchange algebra is given by the universalR-matrix of the quantum groupSL(2) q . This establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group. The method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofSL(2) q . The formulae so derived, which generalize an earlier particular case worked out by Babelon, are remarkably compact and may be entirely written in terms of “q-deformed” factorials and binomial coefficients.

Quantum group structure in the Fock space resolutions of $$\widehat{sl}(n)$$ representations

Abstract: We describe a complex of Wakimoto-type Fock space modules for the affine Kac-Moody algebra\(\widehat{sl}(n)\). The intertwining operators that build the complex are obtained from contour integrals of so-called screening operators. We show that a quantum group structure underlies the algebra of screening operators. This observation greatly facilitates the explicit determination of the intertwiners. We conjecture that the complex provides a resolution of an irreducible highest weight module in terms of Fock spaces.

Representations of affine Kac-Moody algebras, bosonization and resolutions

Abstract: We study boson representations of the affine Kac-Moody algebras and give an explicit description of primary fields and intertwining operators, using vertex operators. We establish the resolution of the irreducible module, consisting of boson representations, and point out the connection with Virasoro algebra. All these give new bosonization procedures for Wess-Zumino-Witten (WZW) models and mathematical backgrounds for the integral representation of correlation functions in WZW models on the plane and on the torus.

A Classical Realization of Quantum Algebras

Abstract: We construct a realization of a deformation of the Lie algebra of a group in terms of the generators of the classical Lie algebra of the group. The construction works for arbitrary (odd) deforming functions and, as a special case, it reproduces the standard quantum deformation of the algebra. For all these functions it gives a co-multiplication, that is, a group homomorphism, and provides an antipode and a co-unit. It therefore promotes any arbitrary deformation into a Hopf algebra.