In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.
@article{bwmeta1.element.doi-10_2478_s11533-006-0014-9,
author = {Dietrich Burde},
title = {Left-symmetric algebras, or pre-Lie algebras in geometry and physics},
journal = {Open Mathematics},
volume = {4},
year = {2006},
pages = {323-357},
zbl = {1151.17301},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0014-9}
}
Dietrich Burde. Left-symmetric algebras, or pre-Lie algebras in geometry and physics. Open Mathematics, Tome 4 (2006) pp. 323-357. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0014-9/
[1] A. d’Andrea and V.G. Kac: “Structure theory of finite conformal algebras”, Selecta Math., Vol. 4, (1998), pp. 377–418. http://dx.doi.org/10.1007/s000290050036 | Zbl 0918.17019
[2] L. Auslander: “Simply transitive groups of affine motions”, Am. J. Math., Vol. 99, (1977), pp. 809–826. | Zbl 0357.22006
[3] C. Bai and D. Meng: “A Lie algebraic approach to Novikov algebras”, J. Geom. Phys. Vol. 45(1–2), (2003), pp. 218–230. http://dx.doi.org/10.1016/S0393-0440(02)00150-X | Zbl 1033.17001
[4] A.A. Balinskii and S.P. Novikov: “Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras”, Sov. Math. Dokl., Vol. 32, (1985), pp. 228–231. | Zbl 0606.58018
[5] B. Bakalov and V. Kac: “Field algebras”, Int. Math. Res. Not., Vol. 3, 2003, pp. 123–159. http://dx.doi.org/10.1155/S1073792803204232 | Zbl 1032.17045
[6] O. Baues: “Left-symmetric algebras for gl(n)”, Trans. Amer. Math. Soc., Vol. 351(7), (1999), pp. 2979–2996. http://dx.doi.org/10.1090/S0002-9947-99-02315-6 | Zbl 1113.17301
[7] Y. Benoist: “Une nilvariété non affine”, J. Differential Geom., Vol. 41, (1995), pp. 21–52.
[8] J.P. Benzécri: Variétés localement affines, Thèse, Princeton Univ., Princeton, N.J., 1955.
[9] R.E. Borcherds: “Vertex algebras, Kac-Moody algebras, and the Monster”, Proc. Nat. Acad. Sci., Vol. 83(10), (1986), pp. 3068–3071. http://dx.doi.org/10.1073/pnas.83.10.3068 | Zbl 0613.17012
[10] N. Boyom: “Sur les structures affines homotopes à zéro des groupes de Lie”, J. Diff. Geom., Vol. 31, (1990), pp. 859–911. | Zbl 0708.53042
[11] D. Burde: “Affine structures on nilmanifolds”, Int. J. Math., Vol. 7, (1996), pp. 599–616. http://dx.doi.org/10.1142/S0129167X96000323 | Zbl 0868.57034
[12] D. Burde: “Simple left-symmetric algebras with solvable Lie algebra”, Manuscripta Math., Vol. 95, (1998), pp. 397–411. http://dx.doi.org/10.1007/s002290050037 | Zbl 0907.17008
[13] D. Burde and K. Dekimpe: “Novikov structures on solvable Lie algebras”, J. Geom. Phys., (2006), to appear. | Zbl 1095.17004
[14] D. Burde and F. Grunewald: “Modules for certain Lie algebras of maximal class”, J. Pure Appl. Algebra, Vol. 99, (1995), pp. 239–254. http://dx.doi.org/10.1016/0022-4049(94)00002-Z | Zbl 0845.17011
[15] D. Burde: “Affine cohomology classes for filiform Lie algebras”, Contemporary Math., Vol. 262, (2000), pp. 159–170. | Zbl 0967.17017
[16] D. Burde: Left-invariant affine structures on nilpotent Lie groups, Habilitation thesis, Düsseldorf, 1999.
[17] D. Burde: “A refinement of Ado’s Theorem”, Archiv Math., Vol. 70, (1998), pp. 118–127. http://dx.doi.org/10.1007/s000130050173 | Zbl 0904.17006
[18] D. Burde: “Estimates on binomial sums of partition functions”, Manuscripta Math., Vol. 103, (2000), pp. 435–446. http://dx.doi.org/10.1007/s002290070002 | Zbl 0983.11059
[19] D. Burde: “Left-invariant affine structures on reductive Lie groups”, J. Algebra, Vol. 181, (1996), pp. 884–902. http://dx.doi.org/10.1006/jabr.1996.0151
[20] A. Cayley: On the Theory of Analytic Forms Called Trees, Collected Mathematical Papers of Arthur Cayley, Vol. 3, Cambridge Univ. Press. Cambridge, 1890, 1890, pp. 242–246.
[21] Y. Carriére, F. Dal’bo and G. Meigniez: “Inexistence de structures affines sur les fibres de Seifert”, Math. Ann., Vol. 296, (1993), pp. 743–753. http://dx.doi.org/10.1007/BF01445134
[22] K.S. Chang, H. Kim and H. Lee: “On radicals of left-symmetric algebra”, Commun. Algebra, Vol. 27(7), (1999), pp. 3161–3175. | Zbl 0932.17002
[23] K.S. Chang, H. Kim and H. Lee: “Radicals of a left-symmetric algebra on a nilpotent Lie group”, Bull. Korean Math. Soc. Vol. 41(2), (2004), pp. 359–369. http://dx.doi.org/10.4134/BKMS.2004.41.2.359 | Zbl 1143.17300
[24] F. Chapoton and M. Livernet: “Pre-Lie algebras and the rooted trees operad”, Intern. Math. Research Notices, Vol. 8, (2001), pp. 395–408. http://dx.doi.org/10.1155/S1073792801000198 | Zbl 1053.17001
[25] A. Connes and D. Kreimer: “Hopf algebras, renormalization and noncommutative geometry”, Comm. Math. Phys., Vol. 199(1), (1998), pp. 203–242. http://dx.doi.org/10.1007/s002200050499 | Zbl 0932.16038
[26] K. Dekimpe and M. Hartl: “Affine structures on 4-step nilpotent Lie algebras” J. Pure Appl. Math., Vol. 129, (1998), pp. 123–134.
[27] K. Dekimpe and W. Malfait: “Affine structures on a class of virtually nilpotent groups”, Topology Appl., Vol. 73, (1996), pp. 97–119. http://dx.doi.org/10.1016/0166-8641(96)00069-7 | Zbl 0878.20023
[28] J. Dixmier and W.G. Lister: “Derivations of nilpotent Lie algebras”, Proc. Amer. Math. Soc., Vol. 8, (1957), pp. 155–158. http://dx.doi.org/10.2307/2032832 | Zbl 0079.04802
[29] J. Dorfmeister: “Quasi-clans”, Abh. Math. Semin. Univ. Hamburg, Vol. 50, (1980), pp. 178–187. http://dx.doi.org/10.1007/BF02941427
[30] A. Dzhumaldil’daev and C. Löfwall: “Trees, free right-symmetric algebras, free Novikov algebras and identities”, Homology Homotopy Appl., Vol. 4(2), (2002), pp. 165–190. | Zbl 1029.17001
[31] A. Dzhumaldil’daev: “N-commutators”, Comment. Math. Helv., Vol. 79(3), (2004), pp. 516–553. | Zbl 1055.17011
[32] A. Dzhumaldil’daev: “Cohomologies and deformations of right-symmetric algebras”, J. Math. Sci., Vol. 93(6), (1999), pp. 836–876. http://dx.doi.org/10.1007/BF02366344 | Zbl 0938.17002
[33] I.B. Frenkel, Y. Huang and J. Lepowsky: “On axiomatic approaches to vertex operator algebras and modules”, Mem. Amer. Math. Soc., Vol. 104(494), (1993), pp. 1–64. | Zbl 0789.17022
[34] I.B. Frenkel, J. Lepowsky and A. Meurman: Vertex operator algebras and the Monster. Pure and Applied Mathematics, Vol. 134, Academic Press, Boston, MA, 1988, pp. 1–508. | Zbl 0674.17001
[35] M. Gerstenhaber: “The cohomology structure of an associative ring”, Ann. Math., Vol. 78, (1963), pp. 267–288. http://dx.doi.org/10.2307/1970343 | Zbl 0131.27302
[36] V. Gichev: “On complete affine structures in Lie groups”, Preprint ArXiv.
[37] W.A. de Graaf: “Constructing faithful matrix representations of Lie algebras”, In: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ACM, New York, pp. 54–59 (electronic). | Zbl 0957.17001
[38] J. Helmstetter: “Radical d’une algèbre symétrique a gauche”, Ann. Inst. Fourier, Vol. 29, (1979), pp. 17–35. | Zbl 0403.16020
[39] N. Jacobson: “A note on automorphisms and derivations of Lie algebras”, Proc. Amer. Math. Soc., Vol. 6, (1955), pp. 281–283. http://dx.doi.org/10.2307/2032356 | Zbl 0064.27002
[40] N. Jacobson: “Schur’s theorem on commutative matrices”, Bull. Amer. Math. Soc., Vol. 50, (1944), pp. 431–436. http://dx.doi.org/10.1090/S0002-9904-1944-08169-X
[41] V. Kac: Vertex algebras for beginners, University Lecture Series, Vol. 10, American Mathematical Society, Providence, 1998, pp. 1–201. | Zbl 0924.17023
[42] H. Kim: “Complete left-invariant affine structures on nilpotent Lie groups”, J. Diff. Geom., Vol. 24, (1986), pp. 373–394. | Zbl 0591.53045
[43] S. Kobayashi and K. Nomizu: Foundations of Differential Geometry, Vols. I and II, Wiley-Interscience Publishers, New York and London, 1969. | Zbl 0175.48504
[44] J.-L. Koszul: “Domaines bornés homogènes et orbites de groupes de transformations affines”, Bull. Soc. Math. France, Vol. 89, (1961), pp. 515–533 | Zbl 0144.34002
[45] D. Kreimer: “New mathematical structures in renormalizable quantum field theories”, Ann. Phys., Vol. 303(1), (2003), pp. 179–202. http://dx.doi.org/10.1016/S0003-4916(02)00023-4 | Zbl 1011.81045
[46] D. Kreimer: “Structures in Feynman Graphs-Hopf Algebras and Symmetries”, Proc. Symp. Pure Math., Vol. 73, (2005), pp. 43–78. | Zbl 1088.81077
[47] N.H. Kuiper: Sur les surfaces localement affines, Colloque de Géometrie différentielle, Strasbourg, 1953, pp. 79–86.
[48] J. Lepowsky and H. Li: “Introduction to Vertex Operator Algebras and Their Representations”, Progr. Math. Vol. 227, (2003), pp. 1–316.
[49] J.P. May: “Geometry of Iterated Moduli Spaces”, Lecture Notes in Math., Vol. 271, 1972.
[50] J. Milnor: “On fundamental groups of complete affinely flat manifolds”, Advances Math., Vol. 25, (1977), pp. 178–187. http://dx.doi.org/10.1016/0001-8708(77)90004-4
[51] A. Mizuhara: “On the radical of a left-symmetric algebra”, Tensor N. S., Vol. 36, (1982), pp. 300–302. | Zbl 0497.17006
[52] A. Mizuhara: “On the radical of a left-symmetric algebra II”, Tensor N. S., Vol. 40, (1983), pp. 221–232. | Zbl 0539.17002
[53] T. Nagano and K. Yagi: “The affine structures on the real two torus”, Osaka J. Math., Vol. 11, (1974), pp. 181–210. | Zbl 0285.53030
[54] A. Nijenhuis: “The graded Lie algebras of an algebra”, Indag. Math., Vol. 29, (1967), pp. 475–486. | Zbl 0153.36201
[55] A. Nijenhuis: “On a class of common properties of some different types of algebras, II”, Nieuw Arch. Wisk. 3, Vol. 17, (1969), pp. 87–108.
[56] M. Nisse: “Structure affine des infranilvariétés et infrasolvariétés”, C. R. Acad. Sci. Paris, Vol. 310, (1990), pp. 667–670. | Zbl 0697.57021
[57] J.M. Osborn: “Novikov algebras”, Nova J. Algebra Geom., Vol. 1(1), (1992), pp. 1–13. | Zbl 0876.17005
[58] J.M. Osborn: “Infinite dimensional Novikov algebras of characteristic 0”, J. Algebra, Vol. 167(1), (1994), pp. 146–167. http://dx.doi.org/10.1006/jabr.1994.1181
[59] B.E. Reed: “Representations of solvable Lie algebras”, Michigan Math. J., Vol. 16, (1969), pp. 227–233. http://dx.doi.org/10.1307/mmj/1029000266
[60] M. Rosellen: “A course in vertex algebra”, Preprint, (2005).
[61] J. Scheuneman: “Affine structures on three-step nilpotent Lie algebras”, Proc. Amer. Math. Soc., Vol. 46, (1974), pp. 451–454. http://dx.doi.org/10.2307/2039945 | Zbl 0291.22010
[62] I. Schur: “Zur Theorie vertauschbarer Matrizen”, J. Reine Angew. Mathematik, Vol. 130, (1905), pp. 66–76. http://dx.doi.org/10.1515/crll.1905.130.66
[63] D. Segal: “The structure of complete left-symmetric algebras”, Math. Ann., Vol. 293, (1992), pp. 569–578. http://dx.doi.org/10.1007/BF01444735 | Zbl 0766.17005
[64] J. Smillie: “An obstruction to the existence of affine structures”, Invent. Math., Vol. 64, (1981), pp. 411–415. http://dx.doi.org/10.1007/BF01389273 | Zbl 0485.57015
[65] W.P. Thurston: Three-dimensional Geometry and Topology, Vol. 1, Princeton Mathematical Series, Vol. 35, Princeton University Press, 1997. | Zbl 0873.57001
[66] E.B. Vinberg: “Convex homogeneous cones”, Transl. Moscow Math. Soc., Vol. 12, (1963), pp. 340–403. | Zbl 0138.43301
[67] E. Zelmanov: “On a class of local translation invariant Lie algebras”, Soviet Math. Dokl., Vol. 35, (1987), pp. 216–218.