Showing papers on "Algebra representation published in 1999"

Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group

Abstract: The Iwahori-Hecke algebra of the symmetric group Cellular algebras The modular representation theory of $\mathcal {H}$ The $q$-Schur algebra The Jantzen sum formula and the blocks of $\mathcal H$ Branching rules, canonical bases and decomposition matrices Appendix A. Finite dimensional algebras over a field Appendix B. Decomposition matrices Appendix C. Elementary divisors of integral Specht modules Index of notation References Index.

Planar algebras, I

Abstract: We introduce a notion of planar algebra, the simplest example of which is a vector space of tensors, closed under planar contractions. A planar algebra with suitable positivity properties produces a finite index subfactor of a II_1 factor, and vice versa.

Lectures on Division Algebras

Abstract: Introduction A division algebra is a central simple algebra Azumaya algebras at the generic point The Brauer group Form of matrices Torsion question Galois extensions Crossed products and cohomology Corestriction Orders and regular domains Ramification Specialization and lifting Lattice methods Brauer Severi varieties Generic division algebra Bibliography.

The Toeplitz algebra of a Hilbert bimodule

Abstract: Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C*-algebra O_X which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebras O_n, and the Cuntz-Krieger algebras O_B. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for the Toeplitz-Cuntz-Krieger algebras of directed graphs, which includes Cuntz's uniqueness theorem for O_\infty.

q-Witt Algebras, q-Lie Algebras, q-Holomorph Structure and Representations ⁄

Abstract: For q generic or q = ", a primitive lth root of 1, q-Witt algebras are described by means of q-divided power algebras. The q-Lie algebras are investi- gated and the q-PBW theorem for universal enveloping algebras of q-Lie algebras is proved. A realization of a class of representations of q-Witt algebras is given. Based on it, the q-holomorph structure for q-Witt algebras is constructed, which interprets the realization in the context of representation theory. 1991 Mathematics Subject Classiflcation: 17B37

Generalized table algebras

Abstract: A table algebra was defined in [1] in order to consider in a uniform way the common properties of conjugacy classes and irreducible characters. Non-commutative table algebras were introduced in [5]. They generalize properties of such well-known objects as coherent and Hecke algebras. Here we extend the main definition of a non-commutative table algebra by letting the ground field be an integral domain. We call these algebrasgeneralized table algebras (GT-algebras, in brief). It is worth mentioning that this class of algebras includes generic Hecke-Iwahori algebras of finite Coxeter groups. We develop the general theory for this type of algebras which includes their representation theory and theory of closed subsets. We also study the properties of primitive integral table algebras.

A quantum analog of the Poincare-Birkhoff-Witt theorem

Abstract: We reduce the basis construction problem for character Hopf algebras to a study of special elements, called “super-letters,” which are defined by Shirshov standard words. It is shown that character Hopf algebras having not more than finitely many “hard” super-letters share some of the properties of universal envelopings of finite-dimensional lie algebras. The background for our proofs is the construction of a filtration such that the associated graded algebra is obtained by iterating the skew polynomials construction, possibly followed with factorization.

Affine sl_p controls the representation theory of the symmetric group and related Hecke algebras

Abstract: In this paper we prove theorems that describe how the representation theory of the affine Hecke algebra of type A and of related algebras such as the group algebra of the symmetric group are controlled by integrable highest weight representations of the characteristic zero affine Lie algebra \hat{sl}_l. In particular we parameterise the representations of these algebras by the nodes of the crystal graph, and give various Hecke theoretic descriptions of the edges. As a consequence we find for each prime p a basis of the integrable representations of \hat{sl}_l which shares many of the remarkable properties, such as positivity, of the global crystal basis/canonical basis of Lusztig and Kashiwara. This {\it $p$-canonical basis} is the usual one when p = 0, and the crystal of the p-canonical basis is always the usual one. The paper is self-contained, and our techniques are elementary (no perverse sheaves or algebraic geometry is invoked).

Diagonal crossed products by duals of quasi-quantum groups

Abstract: A two-sided coaction of a Hopf algebra on an associative algebra ℳ is an algebra map of the form , where (λ,ρ) is a commuting pair of left and right -coactions on ℳ, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra on ℳ by ◃ and ▹, respectively, we define the diagonal crossed product to be the algebra generated by ℳ and with relations given by We give a natural generalization of this construction to the case where is a quasi-Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e. where the coproduct Δ is non-unital). In these cases our diagonal crossed product will still be an associative algebra structure on extending , even though the analogue of an ordinary crossed product in general is not well defined as an associative algebra. Applications of our formalism include the field algebra constructions with quasi-quantum group symmetry given by G. Mack and V. Schomerus [31, 47] as well as the formulation of Hopf spin chains or lattice current algebras based on truncated quantum groups at roots of unity. In the case and λ=ρ=Δ we obtain an explicit definition of the quantum double for quasi-Hopf algebras , which before had been described in the form of an implicit Tannaka–Krein reconstruction procedure by S. Majid [35]. We prove that is itself a (weak) quasi-bialgebra and that any diagonal crossed product naturally admits a two-sided -coaction. In particular, the above-mentioned lattice models always admit the quantum double as a localized cosymmetry, generalizing results of Nill and Szlachanyi [42]. A complete proof that is even a (weak) quasi-triangular quasi-Hopf algebra will be given in a separate paper [27].

Partition Algebras are Cellular

Abstract: The partition algebra P(q) is a generalization both of the Brauer algebra and the Temperley–Lieb algebra for q-state n-site Potts models, underpining their transfer matrix formulation on the arbitrary transverse lattices. We prove that for arbitrary field k and any element q∈ k the partition algebra P(q) is always cellular in the sense of Graham and Lehrer. Thus the representation theory of P(q) can be determined by applying the developed general representation theory on cellular algebras and symmetric groups. Our result also provides an explicit structure of P(q) for arbitrary field and implies the well-known fact that the Brauer algebra D(q) and the Temperley–Lieb algebra TL(q) are cellular.

Post-modern algebra

Abstract: Modern and Post-Modern Algebra. Algebra: The Central Discipline of Mathematics. Sets with Structure and Sets Without Structure. Semigroups and Monoids. GROUP AND QUASIGROUPS. Monoid Actions. Groups and Quasigroups. Symmetry. Loops, Nets and Isotopy. LINEAR ALGEBRA. General Algebra and Linear Algebra. Vector Spaces and Modules. Commutative Algebra. CATEGORIES AND LATTICES. Posets, Monoids and Categories. Limits and Lattices. Adjoint Functors. UNIVERSAL ALGEBRA. Sets with Operations. Varieties. Algebraic Theories. Monads. Index.

Axioms for a vertex algebra and the locality of quantum fields

Abstract: The identities satisfied by two-dimensional chiral quantum fields are studied from the point of view of vertex algebras. The Cauchy-Jacobi identity (or the Borcherds identity) for three mutually local fields is proved and consequently a direct proof of Li's theorem on a local system of vertex operators is provided. Several characterizations of vertex algebras are also discussed.

Cuntz-like algebras

Abstract: The usual crossed product construction which associates to the homeomorphism $T$ of the locally compact space $X$ the C$^*$-algebra $C^*(X,T)$ is extended to the case of a partial local homeomorphism $T$. For example, the Cuntz-Krieger algebras are the C$^*$-algebras of the one-sided Markov shifts. The generalizations of the Cuntz-Krieger algebras (graph algebras, algebras $O_A$ where $A$ is an infinite matrix) which have been introduced recently can also be described as C$^*$-algebras of Markov chains with countably many states. This is useful to obtain such properties of these algebras as nuclearity, simplicity or pure infiniteness. One also gives examples of strong Morita equivalences arising from dynamical systems equivalences.

Uncovering Infinite Symmetries on [p,q] 7-branes: Kac-Moody Algebras and Beyond

Abstract: In a previous paper we explored how conjugacy classes of the modular group classify the symmetry algebras that arise on type IIB [p,q] 7-branes. The Kodaira list of finite Lie algebras completely fills the elliptic classes as well as some parabolic classes. Loop algebras of E_N fill additional parabolic classes, and exotic finite algebras, hyperbolic extensions of E_N and more general indefinite Lie algebras fill the hyperbolic classes. Since they correspond to brane configurations that cannot be made into strict singularities, these non-Kodaira algebras are spectrum generating and organize towers of massive BPS states into representations. The smallest brane configuration with unit monodromy gives rise to the loop algebra \hat{E}_9 which plays a central role in the theory. We elucidate the patterns of enhancement relating E_8, E_9, \hat{E}_9 and E_10. We examine configurations of 24 7-branes relevant to type IIB compactifications on a two-sphere, or F-theory on K3. A particularly symmetric configuration separates the 7-branes into two groups of twelve branes and the massive BPS spectrum is organized by E_10 + E_10.

On the homotopy of simplicial algebras over an operad

Abstract: According to a result of H. Cartan (cf. [5]), the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this paper, we provide a general approach to the construction of such operations in the context of simplicial algebras over an operad. To be precise, we work over a fixed field F, and we consider operads in the category of F-modules. An operad is an algebraic device that specifies a type of algebras. There are operads Com, As, Lie, and Pois, whose algebras are respectively commutative algebras, associative algebras, Lie algebras and Poisson algebras. In general, if P denotes an operad, then we call P-algebras the associated algebras. First, we generalize the notion of a divided power in the context of algebras over an operad. This is done as follows. Recall that the free commutative algebra is given by the formula T (Com, V ) = ⊕n(V )Sn , for V ∈ModF .

Algorithmic Algebra and Number Theory

Abstract: The aim of this article is to describe a computational approach to the study of the arithmetic of modular curves Xo(N) and to give applications of these computations.

On the Groenewold-Van Hove problem for R 2n

Abstract: We discuss the Groenewold–Van Hove problem for R2n, and completely solve it when n=1. We rigorously show that there exists an obstruction to quantizing the Poisson algebra of polynomials on R2n, thereby filling a gap in Groenewold’s original proof. Moreover, when n=1 we determine the largest Lie subalgebras of polynomials which can be consistently quantized, and explicitly construct all their possible quantizations.

A Graphical Parallel Composition Operator for Process Algebras

Abstract: Process algebras are suitable for describing networks of communicating processes. In most process algebras, the description of such networks is achieved using parallel composition operators. Noticing that the parallel composition operators commonly found in process algebras are often limited in expressiveness and/or difficult for novice users, we propose a new parallel operator that allows networks of communicating processes to be described easily, in a simple and well-structured manner. We illustrate on various examples (token-ring network and client-server protocol) the theoretical and practical merits of this operator.

On the affine Temperley-Lieb algebras.

Abstract: The paper describes the cell structure of the affine Temperley–Lieb algebra with respect to a monomial basis. A diagram calculus is constructed for this algebra.

Rational Homotopical Models and Uniqueness

Abstract: Introduction Homotopy theory Differential algebra Complete algebra Three models for spaces Notations Bibliography.