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Affine actions on Lie groups and post-Lie algebra structures. (English) Zbl 06063149
Summary: We introduce post-Lie algebra structures on pairs of Lie algebras (𝔤,𝔫) defined on a fixed vector space V. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtain several results on the existence of post-Lie algebra structures, in terms of the algebraic structure of the two Lie algebras 𝔤 and 𝔫. One result is, for example, that if there exists a post-Lie algebra structure on (𝔤,𝔫), where 𝔤 is nilpotent, then 𝔫 must be solvable. Furthermore special cases and examples are given. This includes a classification of all complex, two-dimensional post-Lie algebras.
MSC:
17Nonassociative rings and algebras
17B30Solvable, nilpotent Lie (super)algebras
17D25Lie-admissible algebras
References:
[1]Bai, C.; Guo, L.; Ni, X.: Nonabelian generalized Lax pairs, the classical Yang – Baxter equation and postlie algebras, Comm. math. Phys. 297, No. 2, 553-596 (2010) · Zbl 1206.17020 · doi:10.1007/s00220-010-0998-7
[2]Baues, O.: Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology 43, No. 4, 903-924 (2004) · Zbl 1059.57022 · doi:10.1016/j.top.2003.11.002
[3]Benoist, Y.: Une nilvariété non affine, C. R. Acad. sci. Paris sér. I math. 315, 983-986 (1992) · Zbl 0776.57010
[4]Burde, D.; Grunewald, F.: Modules for certain Lie algebras of maximal class, J. pure appl. Algebra 99, 239-254 (1995) · Zbl 0845.17011 · doi:10.1016/0022-4049(94)00002-Z
[5]Burde, D.: Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. eur. J. math. 4, No. 3, 323-357 (2006) · Zbl 1151.17301 · doi:10.2478/s11533-006-0014-9
[6]Burde, D.; Beneš, T.: Degenerations of pre-Lie algebras, J. math. Phys. 50, 112102 (2009)
[7]Burde, D.; Dekimpe, K.: Novikov structures on solvable Lie algebras, J. geom. Phys. 56, No. 9, 1837-1855 (2006) · Zbl 1095.17004 · doi:10.1016/j.geomphys.2005.10.010
[8]D. Burde, K. Dekimpe, Post-Lie algebra structures and generalized derivations of semisimple Lie algebras, 2011, preprint.
[9]Burde, D.; Dekimpe, K.; Deschamps, S.: Affine actions on nilpotent Lie groups, Forum math. 21, No. 5, 921-934 (2009) · Zbl 1175.22007 · doi:10.1515/FORUM.2009.045
[10]Burde, D.; Dekimpe, K.; Deschamps, S.: LR-algebras, Contemp. math. 491, 125-140 (2009)
[11]Burde, D.; Dekimpe, K.; Vercammen, K.: Complete LR-structures on solvable Lie algebras, J. group theory 13, No. 5, 703-719 (2010)
[12]Dekimpe, K.: Semi-simple splittings for solvable Lie groups and polynomial structures, Forum math. 12, 77-96 (2000) · Zbl 0946.22009 · doi:10.1515/form.1999.030
[13]Goto, M.: Note on a characterization of solvable Lie algebras, J. sci. Hiroshima univ. Ser. A-I math. 26, No. 1, 1-2 (1962) · Zbl 0142.27602
[14]Jacobson, N.: A note on automorphisms of Lie algebras, Pacific J. Math. 12, No. 1, 303-315 (1962) · Zbl 0109.26201
[15]Loday, J. -L.: Generalized bialgebras and triples of operads, Astrisque 320, 116 (2008) · Zbl 1178.18001 · doi:http://smf.emath.fr/Publications/Asterisque/2008/320/html/smf_ast_320.html
[16]Milnor, J.: On fundamental groups of complete affinely flat manifolds, Adv. math. 25, 178-187 (1977) · Zbl 0364.55001 · doi:10.1016/0001-8708(77)90004-4
[17]Vallette, B.: Homology of generalized partition posets, J. pure appl. Algebra 208, No. 2, 699-725 (2007) · Zbl 1109.18002 · doi:10.1016/j.jpaa.2006.03.012