$Γ$-Conformal Algebras

Abstract

$Γ$-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 $Γ$. To every $Γ$-conformal algebra and a character of $Γ$ 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 ``$Γ$-conformal'' analodue of the general linear algebra) and its analogues, the algebra of pseudodifferential operators on the circle, etc.