Search results for "Nest algebra"

Current Algebras as Hilbert Space Operator Cocycles

1994

Aspects of a generalized representation theory of current algebras in 3 + 1 dimensions axe discussed. Rules for a systematic computation of vacuum expectation values of products of currents are described. Their relation to gauge group actions in bundles of fermionic Fock spaces and to the sesquilinear form approach of Langmann and Ruijsenaars is explained. The regularization for a construction of an operator cocycle representation of the current algebra is explained. An alternative formula for the Schwinger terms defining gauge group extensions is written in terms of Wodzicki residue and Dixmier trace.

The overlap algebra of regular opens

2010

Abstract Overlap algebras are complete lattices enriched with an extra primitive relation, called “overlap”. The new notion of overlap relation satisfies a set of axioms intended to capture, in a positive way, the properties which hold for two elements with non-zero infimum. For each set, its powerset is an example of overlap algebra where two subsets overlap each other when their intersection is inhabited. Moreover, atomic overlap algebras are naturally isomorphic to the powerset of the set of their atoms. Overlap algebras can be seen as particular open (or overt) locales and, from a classical point of view, they essentially coincide with complete Boolean algebras. Contrary to the latter, …

Algebras with involution with linear codimension growth

2006

AbstractWe study the ∗-varieties of associative algebras with involution over a field of characteristic zero which are generated by a finite-dimensional algebra. In this setting we give a list of algebras classifying all such ∗-varieties whose sequence of ∗-codimensions is linearly bounded. Moreover, we exhibit a finite list of algebras to be excluded from the ∗-varieties with such property. As a consequence, we find all possible linearly bounded ∗-codimension sequences.

Almost polynomial growth: Classifying varieties of graded algebras

2015

Let G be a finite group, V a variety of associative G-graded algebras and c (V), n = 1, 2, …, its sequence of graded codimensions. It was recently shown by Valenti that such a sequence is polynomially bounded if and only if V does not contain a finite list of G-graded algebras. The list consists of group algebras of groups of order a prime number, the infinite-dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with suitable gradings. Such algebras generate the only varieties of G-graded algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety is polynomially bounded. In this paper we completely classify all sub…

Lattices of Jordan algebras

2010

AbstractCommutative Jordan algebras play a central part in orthogonal models. The generations of these algebras is studied and applied in deriving lattices of such algebras. These lattices constitute the natural framework for deriving new orthogonal models through factor aggregation and disaggregation.

C*-seminorms on partial *-algebras: an overview

2005

Representations of Affine Kac-Moody Algebras

1989

In the first chapter we explained how simple finite-dimensional Lie algebras can be completely characterized in terms of their Cartan matrices or Dynkin diagrams. The same holds for an arbitrary semisim-ple finite-dimensional Lie algebra. A semisimple Lie algebra is a direct sum of simple ideals which are pairwise orthogonal with respect to the Killing form. It follows that the Cartan matrix of a semisimple Lie algebra decomposes to a block diagonal form, each block representing a simple ideal. Similarly, the Dynkin diagram is a disconnected union of Dynkin diagrams of simple Lie algebras. Next we shall study certain infinite-dimensional Lie algebras which have many similarities with the si…

*-Representations of Partial *-Algebras

2002

This chapter is devoted to *-representations of partial *-algebras. We introduce in Section 7.1 the notions of closed, fully closed, self-adjoint and integrable *-representations. In Section 7.2, the intertwining spaces of two *-representations of a partial *-algebra are defined and investigated, and using them we define the induced extensions of a *-representation. Section 7.3 deals with vector representations for a *-representation of a partial *-algebra, which are the appropriate generalization to a *-representation of the notion of generalized vectors described in Chapter 5. Regular and singular vector representations are defined and characterized by the properties of the commutant, and…