1996/06/11 by Vladimir Hinich, Hinich, Vladimir · 3 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.alg-geom/9606010
openalex publication_date 1996/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To any non-negatively graded dg Lie algebra g over a field k of characteristic zero we assign a functor Σg: art/k → Kan from the category of commutative local artinian k-algebras with the residue field k to the category of Kan simplicial sets. There is a natural homotopy equivalence between Σg and the Deligne groupoid corresponding to g. The main result of the paper claims that the functor Σ commutes up to homotopy with the "total space" functors which assign a dg Lie algebra to a cosimplicial dg Lie algebra and a simplicial set to a cosimplicial simplicial set. This proves a conjecture of Schechtman which implies that if a deformation problem is described ``locally'' by a sheaf of dg Lie algebras g on a topological space X then the global deformation problem is described by the homotopy Lie algebra RΓ(X,g).