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Vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair

Hua-Shin Chang, Hsuan-Yi Liao

Abstract

We investigate vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair $(L,A)$, i.e. a pair of a Lie algebroid $L$ and a Lie subalgebroid $A$ of $L$. The construction of Fedosov dg manifolds involves a choice of a splitting and a connection. We prove that, given any two choices of a splitting and a connection, there exists a unique vertical isomorphism, determined by an iteration formula, between the two associated Fedosov dg manifolds. As an application, we provide an explicit formula for the map $\mathrm{pbw}_2^{-1}\circ \mathrm{pbw}_1$ associated with two Poincaré--Birkhoff--Witt isomorphisms that arise from two choices of a splitting and a connection.

Vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair

Abstract

We investigate vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair , i.e. a pair of a Lie algebroid and a Lie subalgebroid of . The construction of Fedosov dg manifolds involves a choice of a splitting and a connection. We prove that, given any two choices of a splitting and a connection, there exists a unique vertical isomorphism, determined by an iteration formula, between the two associated Fedosov dg manifolds. As an application, we provide an explicit formula for the map associated with two Poincaré--Birkhoff--Witt isomorphisms that arise from two choices of a splitting and a connection.
Paper Structure (15 sections, 20 theorems, 128 equations)

This paper contains 15 sections, 20 theorems, 128 equations.

Table of Contents

  1. Preliminaries
  2. Preliminaries on Lie pairs
  3. Formal graded manifolds and Fedosov manifolds
  4. Fedosov dg manifolds associated with a Lie pair
  5. Koszul-type homotopy equation for $\mathcal{I}$-adic filtration-shifting vertical operators
  6. Topology on the function algebra of a Fedosov manifold
  7. Vertical differential operators
  8. $\mathcal{I}$-adic filtration-shifting vertical operators
  9. Vertical isomorphisms of Fedosov dg manifolds
  10. Vertical automorphisms of a Fedosov manifold
  11. Proof of Theorem \ref{['thm:PhiIteration']}
  12. Applications
  13. Application to geodesic coordinate systems
  14. Application to the $L_\infty$ structure on polydifferential operators
  15. Application to Kapranov dg manifolds

Key Result

Proposition 1.3

Let $(L,A)$ be a Lie pair, and $B=L/A$. Given a splitting $\mathfrak{j}:B \to L$ of the short exact sequence \begin{tikzcd} 0 \ar[r] & A \ar[r, hook, "\ia"'] & \ar[l, bend right, dashed,"\pa"'] L \ar[r, two heads,"\pb"'] & \ar[l, bend right, dashed,"\ib"'] B \ar[r] & 0, \end{tikzcd}and an $L$-conne satisfying for any $f \in R$, $b \in \Gamma(B)$, and $n \in \mathbb{N}$.

Theorems & Definitions (48)

  • Example 1.1
  • Example 1.2
  • Proposition 1.3: MR4271478
  • Remark 1.4
  • Example 1.5
  • Proposition 1.6: MR4150934
  • Proposition 1.7: MR4150934
  • Lemma 2.1
  • proof
  • Remark 2.2
  • ...and 38 more