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On homology of Lie algebras over commutative rings. (English) Zbl 1492.17023

In this paper, the authors compare several definitions of homology for Lie algebras over commutative rings. More specifically, they compare: the homology defined via the \(\mathrm{Tor}\) functor, the one defined via the relative \(\mathrm{Tor}\) functor, the one defined via the Chevalley-Eilenberg complex, the simplicial homology, and the relative simplicial homology. Notice that, in the particular case where the base commutative ring is a field, all these definitions lead to isomorphic results.
In order to better describe their results, let \(\Bbbk\) be a commutative ring and \(\mathfrak g\) be a Lie algebra over \(\Bbbk\). The authors prove that, if \(\mathfrak g\) is a flat \(\Bbbk\)-module, then all of the homologies listed above (with coefficients in the trivial module) are naturally isomorphic (in Theorem 7.3). They also prove that, if \(\Bbbk\) is a principal ideal domain, then the homologies defined via the relative \(\mathrm{Tor}\) functor, the Chevalley-Eilenberg complex and the relative simplicial homology are isomorphic (in Theorems 7.2 and 8.4). Furthermore, the authors provide several examples of isomorphic and non-isomorphic cases, besides a spectral sequence relating the Chevalley-Eilenberg homology to the simplicial one.

MSC:

17B55 Homological methods in Lie (super)algebras
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16S37 Quadratic and Koszul algebras
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References:

[1] Wells Adams, William, An Introduction to Grobner Bases. 3 (1994), American Mathematical Soc. · Zbl 0803.13015
[2] Barr, Michael; Beck, Jon, Acyclic models and triples, (Proceedings of the Conference on Categorical Algebra (1966), Springer), 336-343 · Zbl 0201.35403
[3] Barr, Michael; Beck, Jon, Homology and standard constructions, (Seminar on Triples and Categorical Homology Theory (1969), Springer), 245-335 · Zbl 0176.29003
[4] Barr, Michael; Kennison, John F.; Raphael, Robert, Contractible simplicial objects, Comment. Math. Univ. Carol., 60, 4, 473-495 (2019) · Zbl 1463.18014
[5] Breen, Lawrence, On the functorial homology of Abelian groups, J. Pure Appl. Algebra, 142, 3, 199-237 (1999) · Zbl 0942.20038
[6] Breen, Lawrence; Mikhailov, Roman, Derived functors of nonadditive functors and homotopy theory, Algebraic Geom. Topol., 11, 1, 327-415 (2011) · Zbl 1214.18012
[7] Cartan, Henry; Eilenberg, Samuel, Homological Algebra, vol. 41 (1999), Princeton University Press · Zbl 0933.18001
[8] Cohn, Paul M., A remark on the Birkhoff-Witt theorem, J. Lond. Math. Soc., 1, 1, 197-203 (1963) · Zbl 0109.26202
[9] Dauns, John, Modules and Rings (1994), Cambridge University Press · Zbl 0817.16001
[10] Dixmier, Jc, Homologie des anneaux de Lie, Ann. Sci. Éc. Norm. Supér., 74, 1, 25-83 (1957) · Zbl 0077.04301
[11] Dold, Albrecht; Puppe, Dieter, Homologie nicht-additiver funktoren. anwendungen, Ann. Inst. Fourier, 11, 201-312 (1961) · Zbl 0098.36005
[12] Emmanouil, Ioannis, On pure acyclic complexes, J. Algebra, 465, 190-213 (2016) · Zbl 1350.16008
[13] Farinati, M.; Guccione, Jorge Alberto; Guccione, Juan Jose, The homology of free racks and quandles, Commun. Algebra, 42, 8, 3593-3606 (2014) · Zbl 1300.57017
[14] Fenn, Roger; Rourke, Colin; Sanderson, Brian, The rack space, Trans. Am. Math. Soc., 359, 2, 701-740 (2007) · Zbl 1123.55006
[15] Fenn, Roger; Rourke, Colin; Sanderson, Brian, Trunks and classifying spaces, Appl. Categ. Struct., 3, 4, 321-356 (1995) · Zbl 0853.55021
[16] Fuchs, László, Infinite Abelian Groups (1970), Academic Press · Zbl 0209.05503
[17] Grinberg, Darij, Poincaré-Birkhoff-Witt type results for inclusions of Lie algebras (2011), Citeseer, PhD thesis
[18] Higgins, Philip J., Baer invariants and the Birkhoff-Witt theorem, J. Algebra, 11, 4, 469-482 (1969) · Zbl 0186.06703
[19] Luc, Illusie, Complexe cotangent et déformations I, vol. 239 (2006), Springer
[20] Ivanov, Sergei O.; Romanovskii, Vladislav; Semenov, Andrei, A simple proof of Curtis’ connectivity theorem for Lie powers, Homol. Homotopy Appl., 22, 2, 251-258 (2020) · Zbl 1439.55015
[21] Jean, F., Foncteurs derives de lalgebre symetrique: application au calcul de certains groupes dhomologie fonctorielle des espaces K(B,n) (2002), University of Paris 13, Doctoral thesis, PhD thesis
[22] Lam, Tsit-Yuen, Lectures on Modules and Rings, vol. 189 (2012), Springer Science & Business Media
[23] Loday, Jean-Louis; Pirashvili, Teimuraz, Universal enveloping algebras of Leibniz algebras and (co) homology, Math. Ann., 296, 1, 139-158 (1993) · Zbl 0821.17022
[24] Mac Lane, Saunders, Categories for the Working Mathematician (1971) · Zbl 0232.18001
[25] Mac Lane, Saunders, Homology (2012), Springer Science & Business Media · Zbl 0818.18001
[26] Pirashvili, Teimuraz, Algebra cohomology over a commutative algebra revisited (2003), arXiv preprint · Zbl 1063.19002
[27] Quillen, Daniel, On the (co-)homology of commutative rings, Proc. Symp. Pure Math., 17, 2, 65-87 (1970) · Zbl 0234.18010
[28] Quillen, Daniel G., Homotopical Algebra, vol. 43 (2006), Springer
[29] Reutenauer, Christophe, Free Lie algebras, (Handbook of Algebra, vol. 3 (2003), Elsevier), 887-903 · Zbl 1071.17003
[30] Shirshov, Anatolii Illarionovich, On the representation of Lie rings as associative rings, Usp. Mat. Nauk, 8, 5, 173-175 (1953) · Zbl 0052.03003
[31] Shukla, Umeshachandra, Cohomologie des algebres associatives, Ann. Sci. Éc. Norm. Supér., 78, 2, 163-209 (1961) · Zbl 0228.18005
[32] Szymik, Markus, Quandle cohomology is a Quillen cohomology, Trans. Am. Math. Soc., 371, 8, 5823-5839 (2019) · Zbl 1432.55014
[33] Tierney, Myles; Vogel, Wolfgang, Simplicial resolutions and derived functors, Math. Z., 111, 1, 1-14 (1969) · Zbl 0181.31602
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