[Paper Review] Simultaneous Deformations of Lie Algebroids and Lie Subalgebroids

Xiang Ji|arXiv (Cornell University)|Jul 18, 2012

Advanced Topics in Algebra10 references3 citations

TL;DR

This paper constructs two $L_{ u}$-algebras to govern deformations of Lie algebroids and their Lie subalgebroids, respectively, and combines them into a single $L_{ u}$-algebra to control simultaneous deformations. The key contribution is a unified deformation theory for Lie algebroids and their subalgebroids using higher derived brackets and Maurer-Cartan elements, generalizing classical deformation theory for Lie algebras and extending to foliations, complex structures, and Lie algebroid homomorphisms.

ABSTRACT

The $L_\infty$-algebra is an algebraic structure suitable for describing deformation problems. In this paper we construct one $L_\infty$-algebra, which turns out to be a differential graded Lie algebra, to control the deformations of Lie algebroids and a second one to control the deformations of Lie subalgebroids. We also combine these two $L_\infty$-algebras into one to control the simultaneous deformations of a Lie algebroid and its Lie subalgebroids. The results generalize the deformation theory of Lie algebra and Lie subalgebras. Applications of our results include deformations of foliations, deformations of complex structures and deformations of homomorphisms of Lie algebroids.

Motivation & Objective

• To develop a deformation theory for Lie algebroids using $L_{\infty}$-algebras, generalizing classical DGLA methods.
• To construct a separate $L_{\infty}$-algebra governing deformations of Lie subalgebroids within a given Lie algebroid.
• To unify both deformation problems into a single $L_{\infty}$-algebra framework for simultaneous deformations.
• To apply the framework to geometric structures such as foliations, complex submanifolds, and Lie algebroid homomorphisms.
• To recover and extend known results on Lie algebra homomorphism deformations using supergeometric and derived bracket techniques.

Proposed method

• Uses T. Voronov's method of constructing $L_{\infty}$-algebras via higher derived brackets on graded vector spaces.
• Defines a differential graded Lie algebra (DGLA) on $\mathfrak{X}((A\oplus B)[1])$ to govern Lie algebroid deformations via the Maurer-Cartan equation.
• Applies the same framework to the normal bundle $NS$ and subbundle $E = \operatorname{gr}(\Psi)$ to control deformations of Lie subalgebroids.
• Introduces a V-algebra structure and a flat $L_{\infty}[1]$-algebra $\mathfrak{a}_{X_Q}^P$ to encode deformation data of homomorphisms.
• Uses the identification of Lie algebroid homomorphisms with graphs as Lie subalgebroids in $A \oplus B$ to reduce the problem to subalgebroid deformation.
• Employs the correspondence between deformations and Maurer-Cartan elements in the constructed $L_{\infty}$-algebras to characterize equivalence classes of deformations.

Experimental results

Research questions

• RQ1 How can the deformation theory of Lie algebroids be formulated using $L_{\infty}$-algebras?
• RQ2 What is the $L_{\infty}$-algebraic structure that governs deformations of Lie subalgebroids inside a fixed Lie algebroid?
• RQ3 How can the simultaneous deformation of a Lie algebroid and its Lie subalgebroid be controlled algebraically?
• RQ4 Can the framework be applied to classical geometric deformation problems such as foliations and complex structures?
• RQ5 How do deformations of Lie algebroid homomorphisms relate to the simultaneous deformation of the source, target, and the homomorphism itself?

Key findings

• The $L_{\infty}$-algebra governing deformations of a Lie algebroid is isomorphic to a differential graded Lie algebra, simplifying the structure compared to general $L_{\infty}$-algebras.
• Deformations of a Lie subalgebroid $E \subset A$ are controlled by a flat $L_{\infty}[1]$-algebra $\mathfrak{a}_{X_Q}^P$ constructed from the normal bundle and the graph of the inclusion.
• A bundle map $\tilde{\Psi}: A \to B$ is a deformation of a Lie algebroid homomorphism $\Psi$ if and only if the associated vector field $X_{\sigma,\phi} = h(\tilde{\Psi})$ is a Maurer-Cartan element in $\mathfrak{a}_{X_Q}^P$.
• The simultaneous deformation of Lie algebroid structures on $A$, $B$, and a homomorphism $\Psi: A \to B$ is governed by the $L_{\infty}[1]$-algebra $((\mathfrak{X}(A\oplus B)[1])[1] \oplus \mathfrak{a})_{X_Q}^{P_{I(X_{\sigma,\phi})}}$.
• When $X_Q^2$ is analytic along the normal bundle fibers, the $L_{\infty}[1]$-algebra $\mathfrak{a}_{X_Q}^P$ governs the deformation of the Lie algebroid homomorphism $\Psi$.
• The results recover and generalize the deformation theory of Lie algebra homomorphisms from [22], showing consistency in the special case when algebroids degenerate to Lie algebras.