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Infinitesimal deformations of Lie algebroid pairs

Dadi Ni, Zhuo Chen, Chuangqiang Hu, Maosong Xiang

TL;DR

The paper develops an extended deformation theory for Lie algebroid pairs by enriching the ambient geometric category with local Artinian algebras and constructing governing L∞-algebras that encode infinitesimal deformations.It introduces 𝒜-ringed objects, including 𝒜-ringed Lie algebroids, and identifies small automorphisms and inner automorphisms as the natural equivalences acting on deformations.Two deformation functors are studied: the weak and semistrict deformations of Lie pairs, governed respectively by the extended cubic L∞-algebras 𝔥 and 𝔥₀, and the matched-pair case, where the deformation theory collapses to the dg Lie algebra Ω_A^•(B) and recovers classical results for complex structures and transversely holomorphic foliations.Key results connect Maurer–Cartan theory with deformation classes, showing how gauge equivalence classes of MC elements correspond to isomorphism classes of infinitesimal deformations, and they recover Kodaira–Spencer theory as a special case.

Abstract

We study infinitesimal deformations of Lie algebroid pairs in the category of smooth manifolds enriched with a local Artinian algebra. Given a Lie algebroid pair $(L,A)$, i.e. a Lie algebroid $L$ together with a Lie subalgebroid $A$, we investigate isomorphism classes of infinitesimal deformations of $(L,A)$ modulo automorphisms from exponentials of derivations of $L$ and those from the exponentials of inner derivations of $L$, respectively. For the associated two deformation functors, we find the associated governing $L_\infty$-algebras in the sense of extended deformation theory. Furthermore, when $(L,A)$ is a matched Lie pair, i.e. the quotient $L/A$ is also a Lie subalgebroid of $L$, we investigate isomorphism classes of infinitesimal deformations modulo automorphisms from exponentials of derivations along the normal direction $L/A$. The extended deformation theory of the associated deformation functor recovers the formal deformation theory of complex structures and that of transversely holomorphic foliations.

Infinitesimal deformations of Lie algebroid pairs

TL;DR

The paper develops an extended deformation theory for Lie algebroid pairs by enriching the ambient geometric category with local Artinian algebras and constructing governing L∞-algebras that encode infinitesimal deformations.It introduces 𝒜-ringed objects, including 𝒜-ringed Lie algebroids, and identifies small automorphisms and inner automorphisms as the natural equivalences acting on deformations.Two deformation functors are studied: the weak and semistrict deformations of Lie pairs, governed respectively by the extended cubic L∞-algebras 𝔥 and 𝔥₀, and the matched-pair case, where the deformation theory collapses to the dg Lie algebra Ω_A^•(B) and recovers classical results for complex structures and transversely holomorphic foliations.Key results connect Maurer–Cartan theory with deformation classes, showing how gauge equivalence classes of MC elements correspond to isomorphism classes of infinitesimal deformations, and they recover Kodaira–Spencer theory as a special case.

Abstract

We study infinitesimal deformations of Lie algebroid pairs in the category of smooth manifolds enriched with a local Artinian algebra. Given a Lie algebroid pair , i.e. a Lie algebroid together with a Lie subalgebroid , we investigate isomorphism classes of infinitesimal deformations of modulo automorphisms from exponentials of derivations of and those from the exponentials of inner derivations of , respectively. For the associated two deformation functors, we find the associated governing -algebras in the sense of extended deformation theory. Furthermore, when is a matched Lie pair, i.e. the quotient is also a Lie subalgebroid of , we investigate isomorphism classes of infinitesimal deformations modulo automorphisms from exponentials of derivations along the normal direction . The extended deformation theory of the associated deformation functor recovers the formal deformation theory of complex structures and that of transversely holomorphic foliations.

Paper Structure

This paper contains 16 sections, 18 theorems, 122 equations.

Table of Contents

  1. Lie algebroids enriched with local Artinian algebras
  2. Local Artinian ringed manifolds and vector bundles
  3. Local Artinian ringed Lie algebroids
  4. Infinitesimal deformations of Lie pairs
  5. Infinitesimal deformations and their standard realizations
  6. Weak deformation functors and their standard realizations
  7. Gauge equivalence of Maurer-Cartan elements
  8. Examples
  9. Lie algebra pairs
  10. Extensions of Lie algebroids
  11. Infinitesimal deformations of matched Lie pairs
  12. The deformation functor
  13. Examples
  14. Deformation of complex structures
  15. Transversely holomorphic foliations
  16. ...and 1 more sections

Key Result

Theorem A

The infinitesimal deformation functor ${{\rm wDef}_{{(L,A)}}}$ of weak isomorphisms classes is controlled by the extended cubic $L_\infty$-algebra $\mathfrak{h}$, while ${\rm sDef}_{{(L,A)}}$ of semistrict isomorphism classes is controlled by the $L_\infty$-subalgebra $\mathfrak{h}_0$ of $\mathfrak{

Theorems & Definitions (54)

  • Theorem A
  • Theorem B
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • ...and 44 more