Deformation Theory and Partition Lie Algebras

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.

Final version to appear in Acta Mathematica. 90 pages

References in corpus (5)

• Derived Galois deformation rings
• Enhancing the filtered derived category
• Bredon Homology of Partition Complexes
• Formal moduli problems and formal derived stacks
• Operad bimodules, and composition products on Andre-Quillen filtrations of algebras

Cited by in corpus (7)

• Formal moduli problems and formal derived stacks
• The derived contraction algebra
• The derived deformation theory of a point
• K-theory and polynomial functors
• Presentations of non-commutative deformation rings via $A_{\infty }$-algebras and applications to deformations of Galois representations and pseudorepresentations
• Purely Inseparable Galois theory I: The Fundamental Theorem
• Filtered formal groups, Cartier duality, and derived algebraic geometry