[Paper Review] Deformation Theory and Partition Lie Algebras

Lukas Brantner, Akhil Mathew|arXiv (Cornell University)|Apr 15, 2019

Homotopy and Cohomology in Algebraic Topology41 references12 citations

TL;DR

This paper introduces partition Lie algebras as a characteristic-independent generalization of differential graded Lie algebras, providing a Lie algebraic framework for formal moduli problems in arbitrary characteristic. It establishes an equivalence between formal moduli problems and partition Lie algebras over a field or complete local base, generalizing the Lurie–Pridham correspondence beyond characteristic zero.

ABSTRACT

A theorem of Lurie and Pridham establishes a correspondence between formal moduli problems and differential graded Lie algebras in characteristic zero, thereby formalising a well-known principle in deformation theory. We introduce a variant of differential graded Lie algebras, called partition Lie algebras, in arbitrary characteristic. We then explicitly compute the homotopy groups of free algebras, which parametrise operations. Finally, we prove generalisations of the Lurie-Pridham correspondence classifying formal moduli problems via partition Lie algebras over an arbitrary field, as well as over a complete local base.

Motivation & Objective

• . The paper aims to extend the Lurie–Pridham correspondence—originally valid only in characteristic zero—to positive and mixed characteristic.
• . It addresses the lack of a Lie-theoretic description of derived deformations in finite and mixed characteristic.
• . The objective is to define a new class of homotopical Lie algebras, called partition Lie algebras, that capture the infinitesimal structure of formal moduli problems in arbitrary characteristic.
• . The authors aim to compute the homotopy groups of free partition Lie algebras, which parametrize natural operations analogous to Dyer–Lashof operations.
• . The ultimate goal is to establish a precise correspondence between formal moduli problems and partition Lie algebras over arbitrary fields and complete local rings.

Proposed method

• . The authors define partition Lie algebras using the framework of ∞-categories and monads, leveraging the equivariant topology of partition complexes.
• . They use the theory of hypercoverings and Kan extensions to construct left Kan extensions along subcategories of compact objects.
• . The construction relies on the class S of morphisms inducing surjections on homotopy groups in non-positive degrees.
• . The authors compute the homotopy groups of free partition Lie algebras using the structure of the partition complex |Πn| and its homotopy groups.
• . They apply the Grothendieck construction and nerve arguments to show that certain hypercoverings are colimit diagrams.
• . The key technical tool is the use of filtered colimit-preserving extensions of functors from compact objects to the full category, enabling computation of left Kan extensions.

Experimental results

Research questions

• RQ1 . Can the Lurie–Pridham correspondence between formal moduli problems and differential graded Lie algebras be generalized to positive and mixed characteristic?
• RQ2 . What is the correct homotopical Lie algebraic structure that governs derived deformations in arbitrary characteristic?
• RQ3 . How do the homotopy groups of free partition Lie algebras parametrize natural operations on the homotopy groups of any partition Lie algebra?
• RQ4 . What is the precise relationship between formal moduli problems and partition Lie algebras over a complete local base?
• RQ5 . How can left Kan extensions be computed in the context of ∞-categories of modules and algebras using hypercoverings?

Key findings

• . The paper constructs partition Lie algebras as a characteristic-independent generalization of differential graded Lie algebras.
• . It proves that over any field k, there is an equivalence of ∞-categories between formal moduli problems and partition Lie algebras over k.
• . It establishes a similar equivalence over a complete local base ring, extending the correspondence to arithmetic settings.
• . The homotopy groups of free partition Lie algebras are computed and shown to parametrize Dyer–Lashof-like operations on the homotopy groups of any partition Lie algebra.
• . The authors show that any sifted-colimit-preserving functor on T-algebras is left Kan extended from free algebras on coconnective modules.
• . The construction of left Kan extensions via (F, S)-hypercoverings allows computation of functors even when they do not preserve all geometric realizations.