International Congress of Mathematicians

112 Dimitri Tamarkin2.5.9.Similarly, let A be a *-operad. Then formula 3 defines a Lie-* algebra g A- W ehave natural action of a usual pro-Lie algebra g-p ( M ) on a pro *-Lie algebra gu Mby derivations. The chiral bracket b £ gip, M -, satisfies [6,6] = 0. Therefore, thebracket with 6 defines a differential on gu M . Denote this differential graded *-Liealgebra by VM-2.5.10.To avoid using derived functors, we will slightly modify BM • Recall that M =OJX © X, where N is free. LetV^d(M,...,M;M® V x ) C V cil (M, ...,M;M® V x )be the subset of all operations vanishing under all restrictionsV ch (M, ...,M;M®Vx)-> ^ch( M > • • • ,M,u x ,M,..., M ; M ® V x ).Let defM C BM be the submodule such thatdef M = ®„(^g d (Af, ...,M;M®V x ) ® e„) s " [1 - n].We see that def^f is a *-Lie differential subalgebra of 5M-2.5.11.Proposition 2.4 The functors GM and F^efare canonically isomorphic.2.6. ExampleLet K be a free left F> x -module. Let T X K = K °x i . The symmetric groupSi acts on the Vx -module T % K; let S % K = (T l K) Si be the submodule of invariantsand SK = ®^L 0 S l K. SK is naturally a free commutative F> x -algebra and, hence,SK r := SK ® OJX is a chiral algebra. We will compute the cohomology of theF> x -module deign*. Let SQK = (B < n L 1 S l K. We have:defSK* = ® n (P ch (SoK r [l],..., S 0 K r [l]; SK r ® V x )[l]f".On the other hand, denote by 0 := SK®K. Consider Q as an SK-Vx-moduleof differentials of SK. We have the de Rham differential D : SQK —ï Q. We have athrough mapc„ : P ch (K[l] r ,..., K[l] r , SK[l] r ) s » = F|f (Ü[l] r ,..., Ü[l] r , SK[l] r ) s »Dp ch (s 0 K[iY,...,s 0 K[i] r , SK[iywhere P s £ stands for SFJ-linear chiral operations. Denote by the same letter theinduced mapc, : P ch(K[l] r ,..., K[1Y, SK[lYf- -+ def Sif .