dg Lie algebras

Govern deformation problems:

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koszul duality

Koszul duality patterns have influenced several recent developments in algebraic geometry, ranging from the classification of formal deformations by dg Lie algebras and the unobstructedness of Calabi-Yau varieties to purely inseparable Galois theory and derived Galois deformation rings.
cotangent complex

Relation to dg Lie algebras and deformation theory: deformation theory is controlled by dg Lie algebras.