Operator algebra

About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.

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.

( T * G ) t : A toy model for conformal field theory

Abstract: We study a chiral operator algebra of conformal field theory and quantum deformation of the finite-dimensional Lie group to obtain the definition of (T * G) t and its representation. The closeness of the Kac-Moody algebras, constituting the chiral operator algebra of a typical (and generic) conformal field theory model, namely the WZNW model, and quantum deformation of corresponding finite-dimensional Lie groupG has become more and more evident in recent years [1–5]. This in particular prompts further investigation of the differential geometry of such deformations. The notion of tangent and cotangent bundles is basic in classical differential geometry. It is only natural that the quantum deformations ofTG andT * G are to be introduced alongside those forG itself. Physical ideas could be useful for this goal. Indeed, theT * G can be interpreted as a phase space for a kind of a top, generalizing the usual top associated withG=SO(3). The classical mechanics is a natural language to describe differential geometry, whereas the usual quantization is nothing but the representation theory. In this paper we put corresponding formulas in such a fashion that their deformation becomes almost evident, given the experience in this domain. As a result we get the definition of (T * G) t and its representation (t is the deformation parameter). To make the exposition most simple and formulas transparent we shall work on an example ofG=sl(2) and present results in such a way that the generalizations become evident. We shall stick to generic complex versions, real and especially compact forms requiring some additional consideration, not all of which are self-evident.

Covariant realizations of kappa-deformed space

Abstract: We study a Lie algebra type κ-deformed space with an undeformed rotation algebra and commutative vector-like Dirac derivatives in a covariant way. The space deformation depends on an arbitrary vector. Infinitely many covariant realizations in terms of commuting coordinates of undeformed space and their derivatives are constructed. The corresponding coproducts and star products are found and related in a new way. All covariant realizations are physically equivalent. Specially, a few simple realizations are found and discussed. The scalar fields, invariants and the notion of invariant integration is discussed in the natural realization.