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Homotopy properties of differential Lie modules over curved coalgebras and Koszul duality. (English. Russian original) Zbl 1295.55018
Math. Notes 94, No. 3, 335-350 (2013); translation from Mat. Zametki 94, No. 3, 354-372 (2013).
In the paper under review, the author introduces the notion of a differential Lie module over a curved coalgebra as a generalization of the notion of a differential Lie module over a coalgebra introduced by Smirnov. Then the author describes the basic homotopy and categorical properties of it, and applications of the homotopy technique to studying homotopy properties of differential modules over co-\(B\)-constructions of curved coalgebras Koszul dual to quadratic-scalar algebras.
These quadratic-scalar algebras are not required to be Koszul, and the commutative rings over which they are defined are not required to be fields of characteristic zero. Moreover, the author exemplifies the application of the homotopy constructions to constructing homotopy invariant counterparts of the structure of a differential module over a Clifford algebra and a differential module over an exterior algebra over any commutative unital ring.
The homotopy theory for the homotopy invariant analog of the structure of a differential module over an exterior algebra generalizes the homotopy theory of \(D_\infty\)-differential modules given by the author.

55U30 Duality in applied homological algebra and category theory (aspects of algebraic topology)
16S37 Quadratic and Koszul algebras
Full Text: DOI
[1] Л. Е. Посицельский, “Неоднородная квадратичная двойственность и кривизна”, Функц. анализ и его прил., 27:3 (1993), 57 – 66 · Zbl 0826.16041 · doi:10.1007/BF01768662 · mi.mathnet.ru
[2] S. B. Priddy, “Koszul resolutions”, Trans. Amer. Math. Soc., 152 (1970), 39 – 60 · Zbl 0261.18016 · doi:10.2307/1995637
[3] J. Hirsh, J. Milleś, Curved Koszul Duality Theory, arXiv: 1008.5368v1
[4] V. Ginzburg, M. Kapranov, “Koszul duality for operads”, Duke Math. J., 76:1 (1994), 203 – 272 · Zbl 0855.18006 · doi:10.1215/S0012-7094-94-07608-4
[5] В. А. Смирнов, “Алгебры Ли над операдами и их применение в теории гомотопий”, Изв. РАН. Сер. матем., 62:3 (1998), 121 – 154 · Zbl 0916.55014 · doi:10.1070/im1998v062n03ABEH000198 · mi.mathnet.ru
[6] J. P. May, Simplicial Objects in Algebraic Topology, Van Nostrand Math. Stud., 11, D. Van Nostrand, Princeton, NJ, 1967 · Zbl 0165.26004
[7] В. А. Смирнов, Симплициальные и операдные методы в алгебраической топологии, Факториал, M., 2002
[8] С. В. Лапин, “Дифференциальные возмущения и \(D_\infty \)-дифференциальные модули”, Матем. сб., 192:11 (2001), 55 – 76 · Zbl 1026.55018 · doi:10.1070/SM2001v192n11ABEH000609 · mi.mathnet.ru
[9] С. В. Лапин, “\(D_\infty\)-дифференциальные \(E_\infty\)-алгебры и спектральные последовательности расслоений”, Матем. сб., 198:10 (2007), 3 – 30 · Zbl 1149.55007 · doi:10.1070/SM2007v198n10ABEH003888 · mi.mathnet.ru
[10] С. В. Лапин, “Мультипликативная \(A_\infty\)-структура в спектральных последовательностях расслоений”, Фундамент. и прикл. матем., 14:6 (2008), 141 – 175 · mi.mathnet.ru
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