Fedosov dg manifolds associated with Lie pairs

Abstract

Given any pair (L, A) of Lie algebroids, we construct a differential graded manifold $(L[1]\oplus L/A,Q)$, which we call Fedosov dg manifold. We prove that the homological vector field Q constructed on $L[1]\oplus L/A$ by the Fedosov iteration method arises as a byproduct of the Poincaré–Birkhoff–Witt map established in  [18]. Finally, using the homological perturbation lemma, we establish a quasi-isomorphism of Dolgushev–Fedosov type: the differential graded algebras of functions on the dg manifolds $(A[1],d_A)$ and $(L[1]\oplus L/A,Q)$ are homotopy equivalent.

Appendix A. Homological perturbation

A contraction of a cochain complex $(N,\delta )$ onto a cochain complex (M, d) consists of a pair of cochain maps $\sigma :N\rightarrow M$ and $\tau :M\rightarrow N$ and an endomorphism $h:N\rightarrow N[-1]$ of the graded module N satisfying the following five relations:

$$\begin{aligned}&\sigma \tau ={{\,\mathrm{id}\,}}_M, \qquad \tau \sigma -{{\,\mathrm{id}\,}}_N=h\delta +\delta h, \\&\sigma h=0, \qquad h\tau =0, \qquad h^2=0 . \end{aligned}$$

We symbolize such a contraction by a diagram

If, furthermore, the cochain complexes N and M are filtered and the maps $\sigma$, $\tau$, and h preserve the filtration, the contraction is said to be filtered [9, §12].

A descending filtration

$$\begin{aligned} \cdots \subseteq F_{p+1}N \subseteq F_{p}N \subseteq F_{p-1}N \subseteq \cdots \end{aligned}$$

on a cochain complex N is said to be exhaustive if $N=\bigcup _p F_p N$ and complete if $N=\varprojlim _p \frac{N}{F_p N}$.

A perturbation of the filtered cochain complex

is an operator $\varrho$ of degree $+1$ on N, which raises the filtration degree by 1 (i.e. $\varrho (F_{p} N)\subseteq F_{p+1} N$) and satisfies $(\delta +\varrho )^2=0$ so that $\delta +\varrho$ is a new coboundary operator on N.

We refer the reader to  [15, §1] for a brief history of the following lemma.

Lemma A.1

(Homological Perturbation [6]) Let

be a filtered contraction. Given a perturbation $\varrho$ of the cochain complex $(N,\delta )$, if the filtrations on M and N are exhaustive and complete, then the series

$$\begin{aligned} \breve{\tau }&:=\sum _{k=0}^{\infty }(h\varrho )^k\tau&\breve{h}&:=\sum _{k=0}^{\infty }(h\varrho )^k h=\sum _{k=0}^{\infty }h(\varrho h)^k \\ \breve{\sigma }&:=\sum _{k=0}^{\infty }\sigma (\varrho h)^k&\vartheta&:=\sum _{k=0}^{\infty }\sigma (\varrho h)^k\varrho \tau =\sum _{k=0}^{\infty }\sigma \varrho (h\varrho )^k\tau \end{aligned}$$

converge, $\vartheta$ is a perturbation of the cochain complex (M, d), and

constitutes a new filtered contraction.

Proposition A.2

Under the same hypothesis as in Lemma A.1, the chain map $\breve{\tau }$ is entirely determined by $\tau$, $h\varrho$ and the relation $({{\,\mathrm{id}\,}}-h\varrho )\breve{\tau }=\tau$. Likewise, the homotopy operator $\breve{h}$ is entirely determined by h, $h\varrho$ and the relation $({{\,\mathrm{id}\,}}-h\varrho )\breve{h}=h$.

Proof

Since the filtration on N is complete and $\varrho$ raises the filtration degree by 1 while h preserves it, the geometric series $\sum _{k=0}^\infty (h\varrho )^k$ converges and its sum is the inverse of the operator ${{\,\mathrm{id}\,}}-h\varrho$. The result follows immediately. $\square$