International Congress of Mathematicians

Deformations of Chiral Algebras 1072.2.3. Deformations over a nilpotent X>x-algebraLet F be a nilpotent T>x -algebra with maximal ideal M. We have a notionof F-module and of an F-linear chiral algebra . For any T>x -module M, M F :=1S anM ®Ox E ' F-module.Let M be a chiral algebra. An F-linear unital chiral algebra structure on M Fis called deformation of M over E if the induced structure on M F /M.M F — Mcoincides with the one on M. Denote by GM(E) the set of all isomorphism classesof such deformations.2.3. The functor GM and its representabilityIt is clear that F H> GM(E) is a functor from the category of nilpotent Dxalgebrasto the category of sets. In classical deformation theory one usually has afunctor from the category of (usual) local Arminian (=nilpotent and finitely dimensional)algebras to the category of sets and one tries to represent it by a differentialgraded Lie algebra. In this section we will see that in our situation a natural substitutefor a Lie algebra is a so-called *-Lie algebra in the sense of [1]. More precise,given a *-Lie algebra g, we are going to construct a functor F B from the category ofnilpotent T>x-algebras to the category of sets. In the next section we will show thatthe functor GM is 'pro-representable' in this sense. We will construct a pro-*-Liealgebra dei M (exact meaning will be given below) and an isomorphism of functorsGM and F^f^.2.3.1. *-Lie algebras[1] Let Qi, N be right ©^-modules. SetP*(9i,---,9n,N) := homD x „(ßi M • • • M g n ,i n *N),and F» B (n) := P(g,... ,g;g). It is known that F» B is an operad. A *-Lie algebrastructure on g is by definition a morphism of operads / : lie —t F» B . Let 6 £ lie(2)be the element corresponding to the Lie bracket. We call /(6) £ F» B (2) the *-Liebracket.2.3.2.Let g be a *-Lie algebra and A be a commutative T>x -algebra, introduce avector space g(A) = g ®x> x A. This space is naturally a Lie algebra. Indeed, wehave a *-Lie bracket gij-> Ì2*g- Multiply both parts by A M A:(g m g) ® Vxxx (A m A) -+ i 2 »g ®v xxx (A m A). (*)The left hand side is isomorphic to g(A) ® g(A). The right hand side is isomorphicto g ®T> X (A ®o x -4). Thus, (*) becomes:g(A) ® g(A) -• g ® Vx (A ® 0xA).