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2006 | 4 | 3 | 323-357
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Left-symmetric algebras, or pre-Lie algebras in geometry and physics

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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.
Wydawca
Czasopismo
Rocznik
Tom
4
Numer
3
Strony
323-357
Opis fizyczny
Daty
wydano
2006-09-01
online
2006-09-01
Twórcy
Bibliografia
  • [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
  • [2] L. Auslander: “Simply transitive groups of affine motions”, Am. J. Math., Vol. 99, (1977), pp. 809–826.
  • [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
  • [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.
  • [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
  • [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
  • [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
  • [10] N. Boyom: “Sur les structures affines homotopes à zéro des groupes de Lie”, J. Diff. Geom., Vol. 31, (1990), pp. 859–911.
  • [11] D. Burde: “Affine structures on nilmanifolds”, Int. J. Math., Vol. 7, (1996), pp. 599–616. http://dx.doi.org/10.1142/S0129167X96000323
  • [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
  • [13] D. Burde and K. Dekimpe: “Novikov structures on solvable Lie algebras”, J. Geom. Phys., (2006), to appear.
  • [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
  • [15] D. Burde: “Affine cohomology classes for filiform Lie algebras”, Contemporary Math., Vol. 262, (2000), pp. 159–170.
  • [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
  • [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
  • [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.
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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.
  • [31] A. Dzhumaldil’daev: “N-commutators”, Comment. Math. Helv., Vol. 79(3), (2004), pp. 516–553.
  • [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
  • [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.
  • [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.
  • [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
  • [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).
  • [38] J. Helmstetter: “Radical d’une algèbre symétrique a gauche”, Ann. Inst. Fourier, Vol. 29, (1979), pp. 17–35.
  • [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
  • [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.
  • [42] H. Kim: “Complete left-invariant affine structures on nilpotent Lie groups”, J. Diff. Geom., Vol. 24, (1986), pp. 373–394.
  • [43] S. Kobayashi and K. Nomizu: Foundations of Differential Geometry, Vols. I and II, Wiley-Interscience Publishers, New York and London, 1969.
  • [44] J.-L. Koszul: “Domaines bornés homogènes et orbites de groupes de transformations affines”, Bull. Soc. Math. France, Vol. 89, (1961), pp. 515–533
  • [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
  • [46] D. Kreimer: “Structures in Feynman Graphs-Hopf Algebras and Symmetries”, Proc. Symp. Pure Math., Vol. 73, (2005), pp. 43–78.
  • [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.
  • [52] A. Mizuhara: “On the radical of a left-symmetric algebra II”, Tensor N. S., Vol. 40, (1983), pp. 221–232.
  • [53] T. Nagano and K. Yagi: “The affine structures on the real two torus”, Osaka J. Math., Vol. 11, (1974), pp. 181–210.
  • [54] A. Nijenhuis: “The graded Lie algebras of an algebra”, Indag. Math., Vol. 29, (1967), pp. 475–486.
  • [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.
  • [57] J.M. Osborn: “Novikov algebras”, Nova J. Algebra Geom., Vol. 1(1), (1992), pp. 1–13.
  • [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
  • [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
  • [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
  • [65] W.P. Thurston: Three-dimensional Geometry and Topology, Vol. 1, Princeton Mathematical Series, Vol. 35, Princeton University Press, 1997.
  • [66] E.B. Vinberg: “Convex homogeneous cones”, Transl. Moscow Math. Soc., Vol. 12, (1963), pp. 340–403.
  • [67] E. Zelmanov: “On a class of local translation invariant Lie algebras”, Soviet Math. Dokl., Vol. 35, (1987), pp. 216–218.
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