Dual Formulation of the Lie Algebra S-expansion Procedure

The expansion of a Lie algebra entails finding a new, bigger algebra G, through a series of well-defined steps, from an original Lie algebra g. One incarnation of the method, the so-called S-expansion, involves the use of a finite abelian semigroup S to accomplish this task. In this paper we put forward a dual formulation of the S-expansion method which is based on the dual picture of a Lie algebra given by the Maurer-Cartan forms. The dual version of the method is useful in finding a generalization to the case of a gauge free differential algebra, which in turn is relevant for physical applications in, e.g., Supergravity. It also sheds new light on the puzzling relation between two Chern-Simons Lagrangians for gravity in 2+1 dimensions, namely the Einstein-Hilbert Lagrangian and the one for the so-called "exotic gravity".Comment: 12 pages, no figure

Short Distance Expansion from the Dual Representation of Infinite Dimensional Lie Algebras

We compute the short distance expansion of fields or operators that live in the coadjoint representation of an infinite dimensional Lie algebra by using only properties of the adjoint representation and its dual. We explicitly compute the short distance expansion for the duals of the Virasoro algebra, affine Lie Algebras and the geometrically realized N-extended supersymmetric GR Virasoro algebra.Comment: 19 pages, LaTeX twice, no figure, replacement has corrected Lie algebr

A Three Parameter Hopf Deformation of the Algebra of Feynman-like Diagrams

We construct a three-parameter deformation of the Hopf algebra \LDIAG. This is the algebra that appears in an expansion in terms of Feynman-like diagrams of the {\em product formula} in a simplified version of Quantum Field Theory. This new algebra is a true Hopf deformation which reduces to \LDIAG for some parameter values and to the algebra of Matrix Quasi-Symmetric Functions (\MQS) for others, and thus relates \LDIAG to other Hopf algebras of contemporary physics. Moreover, there is an onto linear mapping preserving products from our algebra to the algebra of Euler-Zagier sums

Γ\Gamma-Conformal Algebras

$\Gamma$-conformal algebra is an axiomatic description of the operator product expansion of chiral fields with simple poles at finitely many points. We classify these algebras and their representations in terms of Lie algebras and their representations with an action of the group $\Gamma$. To every $\Gamma$-conformal algebra and a character of $\Gamma$ we associate a Lie algebra generated by fields with the OPE with simple poles. Examples include twisted affine Kac-Moody algebras, the sin algebra (which is a ``$\Gamma$-conformal'' analodue of the general linear algebra) and its analogues, the algebra of pseudodifferential operators on the circle, etc.Comment: 23 pages, AMSLatex Repotr-no: ITEP-TH-28/9