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Characteristic Classes of Flags of Foliations and Lie Algebra

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Characteristic Classes of Flags of Foliations and Lie Algebra Characteristic classes of flags of foliations and Lie algebra cohomology Anton Khoroshkin ∗ Abstract We prove the conjecture by Feigin, Fuchs and Gelfand describing the Lie algebra cohomology of formal vector fields on an n-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag at the origin. The latter encodes characteristic classes of flags of foliations and was used in the formulation of the local Riemann-Roch Theorem by Feigin and Tsygan. Feigin, Fuchs and Gelfand described the first symmetric power and to do this they had to make use of a fearsomely complicated computation in invariant theory. By the application of degeneration theorems of appropriate Hochschild-Serre spectral sequences we avoid the need to use the methods of FFG, and moreover we are able to describe all the symmetric powers at once. Contents 0 Introduction 2 0.1 Mainresults..................................... ....... 2 0.2 Motivations: formal geometry . ............ 4 0.3 Outlineofthepaper ............................... ........ 5 0.4 Acknowledgments................................. ........ 6 1 Notations and recollections 6 1.1 Lie algebras of formal vector fields and their subalgebras................... 6 1.1.1 Extension by g-valuedfunctions ............................ 6 arXiv:1303.1889v2 [math.KT] 25 Aug 2014 1.1.2 Vector fields preserving foliation structures . ................ 6 1.1.3 Vector fields that are linear in the normal direction to the leaves of a foliation . 7 1.2 Liealgebracohomology .. .. .. .. .. .. .. .. ......... 7 1.3 Weylsuperalgebra ................................ ........ 8 1.3.1 Relativecase .................................. ..... 9 2 Relative chains and truncated Weyl superalgebras 9 2.1 knownresults.................................... ....... 9 2.2 Maintheorems .................................... ...... 10 2.3 Absolutecase .................................... ....... 12 2.4 Characteristic classes of flags of foliations . .................. 13 ∗The research is partially supported by RFBR grants 13-02-00478, 13-01-12401, by ”The National Research University– Higher School of Economics” Academic Fund Program in 2013-2014, research grant 14-01-0124, by Dynasty foundation and Simons-IUM fellowship. 1 3 The core of the proof 15 3.1 Parabolicsubalgebra . .. .. .. .. .. .. .. .. .......... 15 3.2 Degeneration of Hochschild-Serre spectral sequences . .................... 17 3.3 Finalconclusions ................................ ......... 19 4 Particular computations 20 4.1 dimension series for absolute cohomology of W (1,..., 1).................... 20 4.2 Formulas for cocycles in the case of a line (n =1)....................... 22 A Chains on the Lie algebra Wn and gl-invariants 22 A.1 gl-decompositions .................................... 23 A.2 Graphs representing chains on Wm ............................... 23 A.2.1 Chains on WL(m|; g)................................... 24 A.3 formulasforcocycles . .. .. .. .. .. .. .. .. .......... 25 B Relative cohomology of parabolic Lie subalgebra 26 B.1 Notation........................................ ...... 26 B.2 Applications of BGG resolution . ............ 27 B.2.1 Particular case of matrices with two blocks . ............. 29 0 Introduction 0.1 Main results The main result of this paper is a computation of the homology of the Lie algebra Wn of formal vector fields on an n–dimensional space. n ∂ W := { f |f ∈ k[[x ,...,x ]]} n i ∂x i 1 n i i X=1 In the early 70’s B. Feigin, D. Fuchs and I. Gelfand stated a conjectural description of its cohomology with coefficients in symmetric powers of the coadjoint representation. They described applications of this conjecture to formal geometry, Gelfand-Fuchs cohomology and the theory of foliations. This problem was formulated at the famous Gelfand seminar at Moscow State University. In [18] the same authors confirmed their conjecture for the particular case of the first symmetric power of the coadjoint representation. Later on, in 1989, this cohomological conjecture was offered as a proposition1 by B. Feigin and B. Tsygan in [10] and was used in the proof of the local Riemann-Roch theorem. In 2003, V. Dotsenko ([7]) directly computed the cohomology for the linear dual problem in the case n = 1. Namely, he computed the homology of the Lie algebra of polynomial vector fields on the line with coefficients in symmetric powers of the adjoint representation. In this article a uniform method will be presented, which does not involve lengthy computations with invariants and by means of which the cohomologies may be described for all cases of positive n and all symmetric powers at once. The main results can be summarised in the following two statements, proved in Sections 2.3 and 2.2 respectively. Theorem (Theorem 2.3.20, Corollary 2.2.15). For all k > 1 the cohomology of the Lie algebra Wn relative to the subalgebra gln of linear vector fields with coefficients in the k-th symmetric power of the 1without a proof 2 coadjoint representation vanishes everywhere except for degree 2n and the 2nth cohomology coincides with the space of gln-invariants in the (n + k)-th symmetric power of adjoint representation gln: 2n k ∗ n+k gln H (Wn, gln; S Wn ) = [S gln] . (0.1.1) The absolute cohomology has the following description: [Sn+kgl ]gln ⊗ [Λi−2n(gl )]gln , if 2n 6 i 6 n2 + 2n, Hi(W ; SkW ∗)= n n n n 0, otherwise. q Recall that the algebra of gln–invariants in the symmetric algebra S (gln) is isomorphic to the symmetric algebra withq n generators of degrees 1, 2,...,n. Also, the algebra of gln–invariants in the exterior algebra Λ (gln) is isomorphic to the exterior algebra with n odd generators of degrees 1, 3, 5,..., 2n − 1. This is explained for example in [31, 12]. In particular, we can identify the 2nth coho- 2n k ∗ mology H (Wn; S Wn ) of order 2n with the subspace of polynomials of degree n + k in the polynomial algebra k[x1,x2,...,xn] subject to the condition that the generator xi has degree i. Formulas for the generating cocycles of (0.1.1) are given in Section §A.3. At the same time we prove another homological conjecture initially formulated in formal geometry and in foliation theory (see e.g. [9]): we compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag of foliations. More precisely, let us fix a collection of natural numbers n = (n0,...,nk) with |n| = n0 +. .+nk, and fix a sequence of trivial embedded foliations {F1, F2,..., Fk} |n| |n| in R such that the foliation Fi+1 has codimension ni in Fi (n0 is the codimension of F1 in R . The Lie algebra W (n0,...,nk) consists of formal vector fields that infinitesimally preserve all foliations near the origin (see section 1.1 for details). That is f1,...,fn ∈ R[[x1,...,x ]], |n| 0 |n| ∂ fn0+1,...,fn0+n1 ∈ R[[xn0+1,...,x|n|]], W (n0,...,nk) := fi ∂xi fn0+n1+1,...,fn0+n1+n2 ∈ R[[xn0+n1+1,...,x|n|]], i=1 X . Consider the maximal reductive subalgebra in the space of linear vector fields that preserve the flag gl mentioned above. This subalgebra is isomorphic to the direcq t sum of matrix algebras n0 ⊕ . ⊕ gl gl gl nk . Let In0,...,nk be the ideal in the symmetric algebra S ( n0 ⊕ . ⊕ nk ) generated by the sum k n0+...+nr+1 ⊕ S (gln ⊕ . ⊕ gln ). We state the description of the relative cohomology in terms of the r=0 0 r quotient algebra: Theorem (Cor.2.2.19 below). The cohomology of the Lie algebra W (n0,...,nk) relative to subalgebra gl ⊕. .⊕gl with trivial coefficients are different from zero only in even degrees and coincides with the n0 nk q gl gl gl gl algebra of n0 ⊕ . ⊕ nk invariants in the factor-algebra of the symmetric algebra S ( n0 ⊕ . ⊕ nk ) by the ideal In0,...,nk . In particular, for all i we have: 0, if i = 2k + 1, i gln ⊕...⊕gln H (W (n0,...,nk), gl ⊕ . ⊕ gl ; k)= Sk(gl ⊕...⊕gl ) 0 k n0 nk n0 nk , if i = 2k. In0,...,nk Moreover, the algebra of gl invariants has canonical description as a ring of symmetric functions: q gl ⊕...⊕gl gl gl n0 nk k S ( n0 ⊕ . ⊕ nk ) = [Ψ01,..., Ψ0n0 ; . ;Ψk1,..., Ψknk ] . Namely, the aforementioned ring is the free commutative algebra generated by the set of generators {Ψij}, index i ranges from 0 to k and j ranges from 1 to ni. The degree of the generator Ψij coincides with 3 the second index j. The intersection of the ideal In0,...,nk with the subalgebra of invariants is generated α01 α0n0 αr1 αrnr r ni by monomials (Ψ01 . Ψ0n0 ) . (Ψr1 . Ψrnr ) such that ( i=0 j=1 αij) >n0 + · · · + nr, where the index r ranges from 0 to k as above. P P The answer for the absolute cohomology of the Lie algebra W (n0,...,nk) may be complicated for particular choice of integer numbers (n0,...,nk), however there exists a straitforward procedure on how one should complute this absolute cohomology. We refer these discussions to Section 2.3 and in particular to Theorem 2.3.21. The particular case n = (1, 1,..., 1) where all dimensions are computed in terms of Catalan numbers is considered in Section 4.1. 0.2 Motivations: formal geometry In the early 1970s I. M. Gelfand presented the description of characteristic classes of a manifold of dimen- sion n using the Lie algebra cohomology of the infinite-dimensional Lie algebra of formal vector fields Wn. (See e.g. [16] written after presentations at the Gelfands famous seminar, and an overview after ICM in Nice [15]; more details can be found in [12]ch.3,[16],[1],[3].) The core construction is called formal geometry or Gelfand-Fuchs cohomology and looks as follows. To any complex smooth manifold M of dimension n we assign the infinite-dimensional manifold M coor of formal coordinates on M. Formally, a point of M coor is a pair: a point p ∈ M together with a system of formal coordinates in a neighbourhood of p.
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