2020/03/12 by Anton Khoroshkin, Khoroshkin, Anton, Pedro Tamaroff +1 · 1 citation
Mathematics · Medicine · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Cancer Treatment and Pharmacology #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2003.06055
openalex publication_date 2020/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define derived Poincar 'e--Birkhoff--Witt maps of dg operads or derived\nPBW maps, for short, which extend the definition of PBW maps between operads of\nV.~Dotsenko and the second author in 1804.06485, with the purpose of studying\nthe universal enveloping algebra of dg Lie algebras as a functor on the\nhomotopy category. Our main result shows that the map from the homotopy Lie\noperad to the homotopy associative operad is derived PBW, which gives us an\namenable description of the homology of the universal envelope of an\nL_\∞-algebra in the sense of Lada--Markl. We deduce from this several\nknown results involving universal envelopes of L_\∞-algebras of V.\nBaranovsky and J. Moreno-Fern 'andez, and extend D. Quillen's classical\nquasi-isomorphism mathcal C longrightarrow BU from dg Lie algebras to\nL_\∞-algebras; this confirms a conjecture of J. Moreno-Fern 'andez.\n