A canonical semi-classical star product (original) (raw)

On deformation of associative algebras and graph homology

2007

Deformation theory of associative algebras and in particular of Poisson algebras is reviewed. The role of an "almost contraction" leading to a canonical solution of the corresponding Maurer-Cartan equation is noted. This role is reminiscent of the homotopical perturbation lemma, with the infinitesimal deformation cocycle as "initiator". Applied to star-products, we show how Moyal's formula can be obtained using such an almost contraction and conjecture that the "merger operation" provides a canonical solution at least in the case of linear Poisson structures.

A particular type of non-associative algebras and graph theory

2011

Evolution algebras have many connections with other mathematical fields, like group theory, stochastics processes, dynamical systems and other related ones. The main goal of this paper is to introduce a novel non-usual research on Discrete Mathematics regarding the use of graphs to solve some open problems related to the theory of graphicable algebras, which constitute a subset of those algebras. We show as many our advances in this field as other non solved problems to be tackled in future.

Graph operations and Lie algebras

International Journal of Computer Mathematics, 2013

>IJH=?J This paper deals with several operations on graphs and combinatorial structures linking them with their associated Lie algebras. More concretely, our main goal is to obtain some criteria to determine when there exists a Lie algebra associated with a combinatorial structure arising from those operations. Additionally, we show an algorithmic method for one of those operations.

Applications of the Gauge-invariant uniqueness theorem for graph algebras

Bulletin of the Australian Mathematical Society, 2002

We give applications of the gauge-invariant uniqueness theorem, which states that the Cuntz-Krieger algebras of directed graphs are characterised by the existence of a canonical action of . We classify theC*-algebras of higher order graphs, identify theC*-algebras of cartesian product graphs with a certain fixed point algebra and investigate a relation called elementary shift equivalence on graphs and its effect on the associated graphC*-algebras.

On deformation theory and graph homology

Arxiv preprint math/0507077, 2005

Deformation theory of associative algebras and in particular of Poisson algebras is reviewed. The role of an "almost contraction" leading to a canonical solution of the corresponding Maurer-Cartan equation is noted. This role is reminiscent of the homotopical perturbation lemma, with the infinitesimal deformation cocycle as "initiator". Applied to star-products, we show how Moyal's formula can be obtained using such an almost contraction and conjecture that the "merger operation" provides a canonical solution at least in the case of linear Poisson structures.

algebras of labelled graphs III—-theory computations

Ergodic Theory and Dynamical Systems, 2015

In this paper we give a formula for the KKK -theory of the CastC^{\ast }Cast -algebra of a weakly left-resolving labelled space. This is done by realizing the CastC^{\ast }Cast -algebra of a weakly left-resolving labelled space as the Cuntz–Pimsner algebra of a CastC^{\ast }Cast -correspondence. As a corollary, we obtain a gauge-invariant uniqueness theorem for the CastC^{\ast }Cast -algebra of any weakly left-resolving labelled space. In order to achieve this, we must modify the definition of the CastC^{\ast }Cast -algebra of a weakly left-resolving labelled space. We also establish strong connections between the various classes of CastC^{\ast }Cast -algebras that are associated with shift spaces and labelled graph algebras. Hence, by computing the KKK -theory of a labelled graph algebra, we are providing a common framework for computing the KKK -theory of graph algebras, ultragraph algebras, Exel–Laca algebras, Matsumoto algebras and the CastC^{\ast }Cast -algebras of Carlsen. We provide an inductive limit approach for co...

Structural properties of the graph algebra

Journal of Pure and Applied Algebra, 2008

The algebras Q n describe the relationship between the roots and coefficients of a non-commutative polynomial. I. Gelfand, S. Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this paper we find the Hilbert series of the class of algebras corresponding to the graph K 3 and show that this algebra is Koszul.

On deformation theory of quantum vertex algebras

Eprint Arxiv Hep Th 0508225, 2005

We study an algebraic deformation problem which captures the data of the general deformation problem for a quantum vertex algebra. We derive a system of coupled equations which is the counterpart of the Maurer-Cartan equation on the usual Hochschild complex of an assocative algebra. We show that this system of equations results from an action principle. This might be the starting point for a perturbative treatment of the deformation problem of quantum vertex algebras. Our action generalizes the action of the Kodaira-Spencer theory of gravity and might therefore also be of relevance for applications in string theory.

Algebraic structures on graph associahedra

Journal of the London Mathematical Society

M. Carr and S. Devadoss introduced in [7] the notion of tubing on a finite simple graph Γ, in the context of configuration spaces on the Hilbert plane. To any finite simple graph Γ they associated a finite partially ordered set, whose elements are the tubings of Γ and whose geometric realization is a convex polytope KΓ, the graph-associahedron. For the complete graphs they recovered permutahedra, for linear graphs they got Stasheff's associahedra, while for simple graph they obtained the standard simplexes. The goal of the present work is to give an algebraic description of graph associahedra. We introduce a substitution operation on tubings, which allows us to describe the set of faces of graph-associahedra as a free object, spanned by the set of all connected simple graphs, under operations given via connected subgraphs. The boundary maps of graphassociahedra defines natural derivations in this context. Along the way, we introduce a topological interpretation of the graph tubings and our new operations. In the last section, we show that substitution of tubings may be understood in the context of M. Batanin and M. Markl's operadic categories.

Realization of algebras with the help of star-products

2004

We present a closed formula for a family of star-products by replacing the partial derivatives in the Moyal-Weyl formula with commuting vector fields. We show how to reproduce algebra relations on commutative spaces with these star-products and give some physically interesting examples of that procedure.