International Congress of Mathematicians

3.1.5.Deformations of Chiral Algebras 115iLet Ai be right F> x -modules. Write A ± ® A 2 := (A[ ®A l 2) r ; Pi(Ai,A 2 ;A 3 ) :=hom(.4i ® A 2 ,A 3 ).We have a mapWe have (.4 ® B) = i* 2 (B M C);ii 2 *(A® B) -+i 2 *A®p* 2 (B l ).c : P r .(Ai,A 2 ;B) ® P,(B, C; D) -+ P»(Ai,A 2 ® C; D)1defined as follows. Let u : Ai M A 2 —¥ Ì2*B and m : B ® C —¥ D. Putc(u,v) : AiM(A 2 ® C) = (-4 1 H.4 2 )cgi_ P ;(C') -^ i 2 *B®p* 2 (C l ) ^ i 2 .(B ® C) -+ i 2 .D.3.1.6.Denoteie = c(0,U ft ) G P*(HM,H M ® H M ;H M ).1 1Let T : HM ® HM —ï H M ® HM be the action of symmetric group and let e Tthe composition with T. Let / G P*(HM,HM® HM', HM) be defined by:beHM E3 (HM® HM) -^ HM E3 HM ~^ ì*HM-Proposition 3.3 We have f = e + e T .3.1.7.In other words, the cup product and the bracket satisfy the Leibnitz identity.We see that HM has a pro-*-Lie bracket, (HM)'[1] has a commutative Vxalgebrastructure, and these structures satisfy the Leibnitz identity. Call this structurea c-Gerstenhaber algebra structure. Thus, our findings can be summarized asfollows.Theorem 3.4 The cohomology of the deformation pro-*-Lie algebra of a chiralalgebra is naturally a pro-c-Gerstenhaber algebra.3.2. Example M = (SK) rWe come back to our example 2.6.. For simplicity assume K is finitely generatedfree Vx -module. We have seen in this case that(H M ) l [-l] ~ (Bi A* K v ® S K [^i] ~ S(K V [^1] e K).Proposition 3.5 The cup product on HM coincides with the natural one on thesymmetric power algebra.