Derived brackets. (English) Zbl 1055.17016

Kosmann-Schwarzbach, Yvette

Lett. Math. Phys. 69, Spec. Iss., 61-87 (2004).

The author describes several differential geometric settings that illustrate the notion of derived bracket introduced by herself [Ann. Inst. Fourier 46, No. 5, 1243–1274 (1996; Zbl 0858.17027)]. We recall that, given a graded differential Lie algebra \((V,[\cdot,\cdot],D)\) with the bracket of degree \(n\), the corresponding derived bracket is the bilinear mapping \([\cdot,\cdot]_{(D)}: V\times V\to V\) defined by \([a,b]_{(D)}=(-1)^{n+| a|+1}=[Da,b]\) for all \(a,b\in V\), where \(| a|\) stands for the degree of \(a\).
The paper is written in a clear style, and includes many instructive historical remarks, as well as a rather extensive list of references.

Reviewer: Daniel Beltiţă (Bucureşti)

MSC:

17B70	Graded Lie (super)algebras
53D17	Poisson manifolds; Poisson groupoids and algebroids
17B66	Lie algebras of vector fields and related (super) algebras
17B63	Poisson algebras
17D99	Other nonassociative rings and algebras
58A50	Supermanifolds and graded manifolds

Keywords:

derived bracket; Courant bracket; Loday-Leibniz algebra

Citations:

Zbl 0858.17027

References:

[1]	Alekseev, A. and Xu, P.: Derived brackets and Courant algebroids, in preparation.
[2]	Alexandrov, M., Schwarz, A., Zaboronsky, O. and Kontsevich, M.: The geometry of the master equation and topological quantum field theory, Internat.J.Modern Phys.A 12 (7) (1997), 1405-429. · Zbl 1073.81655
[3]	Bangoura, M. and Kosmann-Schwarzbach, Y.: The double of a Jacobian quasi-bialgebra, Lett.Math.Phys. 281993), 13-29. · Zbl 0796.17008
[4]	Batalin, I. and Marnelius, R.: Dualities between Poisson brackets and antibrackets, Internat.J.Modern Phys.A 14 (32) (1999), 5049-5073. · Zbl 1125.81306
[5]	Bursztyn, H. and Crainic, M.: Dirac structures, moment maps and quasi-Poisson manifolds, In:J. E. Marsden and T. S. Ratiu (eds), The Breadth of Symplectic and Poisson Geometry, Progr. in Math., Birkh ”auser, Boston, 2004. · Zbl 1079.53123
[6]	Buttin, C.: Les d ’erivations des champs de tenseurs et l ’invariant différentiel de Schouten, C.R.Acad.Sci.Paris A 2691969), 87-89. · Zbl 0187.43904
[7]	Buttin, C.: Th ’eorie des op ’erateurs différentiels gradu ’es sur les formes différentielles, Bull.Soc.Math.France 1021974), 49-73. · Zbl 0285.58014
[8]	Cabras, A. and Vinogradov, A. M.: Extensions of the Poisson bracket to differential forms and multi-vector elds, J.Geom.Phys. 91992), 75-100. · Zbl 0748.58008
[9]	Cartan, H.: Notion d ’algebre différentielle;applications aux groupes de Lie et aux vari ’et ’es ouopere un groupe de Lie, Colloque de Topologie, tenu a Bruxelles, 5-8 juin 1950, CBRM, Georges Thone, Liege, 1950, pp. 15-27.
[10]	Cartier, P.: Some fundamental techniques in the theory of integrable systems, In:O. Babelon, P. Cartier and Y. Kosmann-Schwarzbach (eds.), Lectures on Integrable Systems, World Scientific, Singapore, 1994, pp. 1-41.
[11]	Courant, T. J.: Dirac manifolds, Trans.Amer.Math.Soc. 3191990), 631-661. · Zbl 0850.70212
[12]	Courant, T. J. and Weinstein, A.: Beyond Poisson structures, In: P. Dazord and N. Desolneux-Moulis (eds), Actions hamiltoniennes de groupes.Troisiemne th ’eoreme de Lie (Lyon, 1986), Travaux en Cours 27, Hermann, Paris, 1988. · Zbl 0698.58020
[13]	Daletskii, Yu. L. and Kushnirevitch, V. A.: Formal differential geometry and NambuTakhtajan algebra, Banach Center Publ. 401997), 293-302. · Zbl 0883.17029
[14]	Dorfman, I. Ya.: Dirac structures of integrable evolution equations, Physics Lett.A 1251987), 240-246.
[15]	Dorfman, I. Ya.: Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, New York, 1993. · Zbl 0717.58026
[16]	Dubois-Violette, M. and Michor, P.: More on the Fr ”olicher-Nijenhuis bracket in non commutative differential geometry, J.Pure Appl.Algebra 1211997), 107-135. · Zbl 0889.58011
[17]	Fr”olicher, A. and Nijenhuis, A.: Theory of vector-valued differential forms, Part I, Indag.Math. 181956), 338-359. · Zbl 0079.37502
[18]	Gel ’fand, I. M., Daletskii, Yu. L. and Tsygan, B. L.: On a variant of noncommutative differential geometry, Dokl.Akad.Nauk SSSR 308 (6) (1989), 1293-1297;Soviet Math.Dokl. 40 (2) (1990), 422-426. (See also Seminar on Supermanifolds, Report no. 16, Stockholm, 1987.)
[19]	Gel ’fand, I. M. and Dorfman, I. Ya.: Schouten bracket and Hamiltonian operators, Funkt.Anal.i Prilozhen. 14 (3) (1980), 71-74;Funct.Anal.Appl. 141981), 223-226. · Zbl 0444.58010
[20]	Grabowski, J.: Extension of Poisson brackets to differential forms, In:H.-D. Doebner, W. Scherer and C. Schulte, (eds), Group 21, Vol. 2, World Scientific, Singapore, 1997; Z-graded extensions of Poisson brackets, Rep.Math.Phys. 91997), 1-27. · Zbl 0872.58029
[21]	Grabowski, J. and Urba ’nski, P.: Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids, Ann.Global Anal.Geom. 151997), 447-486. · Zbl 0973.58006
[22]	Hitchin, N.: Generalized Calabi-Yau manifolds, Quart.J.Math. 542003), 281-305. · Zbl 1076.32019
[23]	Klim !c ’ýk, C. and Strobl, T.: WZW-Poisson manifolds, J.Geom.Phys. 432002), 341-344. · Zbl 1027.70023
[24]	Kosmann-Schwarzbach, Y.: Jacobian quasi-bialgebras and quasi-Poisson Lie groups, Contemp.Math. 1321992), 459-489. · Zbl 0847.17020
[25]	Kosmann-Schwarzbach, Y.: Exact Gerstenhaber algebras and Lie bialgebroids, Acta.Appl.Math. 411995), 153-165. · Zbl 0837.17014
[26]	Kosmann-Schwarzbach, Y.: From Poisson algebras to Gerstenhaber algebras, Ann.Inst.Fourier (Grenoble )461996), 1241-1272. · Zbl 0858.17027
[27]	Kosmann-Schwarzbach, Y.: Derived brackets and the gauge algebra of closed string eld theory, In:H.-D. Doebner and V. K. Dobrev (eds), Quantum Group Symposium at GROUP 21(Goslar,1996 ), Heron Press, Sofia, 1997, pp. 53-61.
[28]	Kosmann-Schwarzbach, Y.: Odd and even Poisson brackets, In:H.-D. Doebner, V. K. Dobrev, J.-D. Hennig and W. L ”ucke (eds.), Quantum Theory and Symmetries, World Scientific, Singapore, 2000, pp. 565-571. · Zbl 0981.53080
[29]	Kosmann-Schwarzbach, Y.: Quasi, twisted and all that... in Poisson geometry and Lie algebroid theory, In:J. E. Marsden and T. S. Ratiu (eds), The Breadth of Symplectic and Poisson Geometry, Progr. in Math., Birkh ”auser, Boston, 2004. · Zbl 1079.53126
[30]	Kosmann-Schwarzbach, Y. and Magri, F.: Poisson-Nijenhuis structures, Ann.Inst.Henri Poincar ’e A 531990), 35-81. · Zbl 0707.58048
[31]	Kostant, B. and Sternberg, S.: Symplectic reduction, BRS cohomology and infinitedimensional Clifford algebras, Ann.Phys. (N.Y.) 1761987), 49-113. · Zbl 0642.17003
[32]	Koszul, J.-L.: Crochet de Schouten-Nijenhuis et cohomologie, Ast ’erisque, hors s ’erie, (1985), 257-271.
[33]	Koszul, J.-L.: Personal communication (1994)of unpublished notes (1990).
[34]	Krasil ’shchik, I.: Supercanonical algebras and Schouten brackets, Mat.Zam. 49 (1) (1991), 70-76;Math.Notes 49 (1-2) (1991), 50-54.
[35]	Kumpera, A. and Spencer, D.: Lie Equations, Volume I,General Theory, Princeton University Press, 1972. · Zbl 0258.58015
[36]	Lecomte, P. and Roger, C.: Modules et cohomologie des bigebres de Lie, C.R.Acad.Sci.Paris, S ’er. I 3101990), 405-410.
[37]	Lichnerowicz, A.: Les vari ’et ’es de Poisson et leurs algebres de Lie associ ’ees, J.Differential Geom. 121977), 253-300. · Zbl 0405.53024
[38]	Liu, Z.-J., Weinstein, A. and Xu, P.: Manin triples for Lie bialgebroids, J.Differential Geom. 451997), 547-574. · Zbl 0885.58030
[39]	Loday, J.-L.: Une version non commutative des algebres de Lie:les algebres de Leibniz, Enseig.Math. 391993), 613-646.
[40]	Mackenzie, K. C. H. and Xu, P.: Lie bialgebroids and Poisson groupoids, Duke Math.J. 731994), 415-452. · Zbl 0844.22005
[41]	Michor, P. W.: A generalization of Hamiltonian dynamics, J.Geom.Phys. 2 (2) (1985), 67-82. · Zbl 0587.58004
[42]	Nijenhuis, A.: Jacobi-type identities for bilinear differential concomitants of certain tensor elds, Indag.Math. 171955), 390-403. · Zbl 0068.15001
[43]	Nijenhuis, A.: The graded Lie algebras of an algebra, Indag.Math. 291967), 475-486. · Zbl 0153.36201
[44]	Nijenhuis, A. and Richardson, R.: Deformations of Lie algebra structures, J.Math.Mech. 1711967), 89-106. · Zbl 0166.30202
[45]	Ouzilou, R.: Hamiltonian actions on Poisson manifolds, In:A. Crumeyrolle and J. Grifone (eds), Symplectic Geometry, Pitman, London, 1983, pp. 172-183. · Zbl 0514.58010
[46]	Park, J. S.: Topological open p-branes, In:K. Fukaya, Y.-G. Oh, K. Ono and G. Tian (eds.), Symplectic Geometry and Mirror Symmetry (Seoul,2000 ), World Scientific, Singapore, 2001, pp. 311-384. · Zbl 1024.81043
[47]	Roger, C.: Algebres de Lie gradu ’ees et quanti cation, In:P. Donato, C. Duval, J. Elhadad and G. M. Tuynman (eds), Symplectic Geometry and Mathematical Physics, Progr. in Math. 99, Birkh ”auser, Basel, 1991, pp. 374-421. · Zbl 0748.17028
[48]	Roytenberg, D.: Courant algebroids, derived brackets and even symplectic supermanifolds, PhD thesis, U. C. Berkeley, 1999;mathDG/9910078.
[49]	Roytenberg, D.: Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett.Math.Phys. 612002), 123-137. · Zbl 1027.53104
[50]	Roytenberg, D.: On the structure of graded symplectic supermanifolds and Courant algebroids, In: T. Voronov (ed.), Quantization,Poisson Brackets and Beyond, Contemp. Math. 315, Amer. Math. Soc., Providence, 2002, pp. 169-185. · Zbl 1036.53057
[51]	Schouten, J. A.: ”Uber Differentialkomitanten zweier kontravarianter Gr ”ossen, Proc.Ned.Akad.Wet.Amsterdam 431940), 449-452. · Zbl 0023.17002
[52]	!Severa, P. and Weinstein, A.: Poisson geometry with a 3-form background, Progr.Theor.Phys.Suppl. 1442001), 145-154.
[53]	Tulczjyew, W. M.: The graded Lie algebra of multivector elds and the generalized Lie derivative of forms, Bull.Acad.Pol.Sci. 221974), 937-942.
[54]	Vaintrob, A.: Lie algebroids and homological vector elds, Uspekhi Mat.Nauk 52 (2) (1997), 161-162;Russian Math.Surv. 52 (2) (1997), 428-429. · Zbl 0955.58017
[55]	Vinogradov, A. M.: Uni cation of the Schouten and Nijenhuis brackets, cohomology, and superdifferential operators, Mat.Zam. 47 (6) (1990), 138-140 [not translated in Math.Notes ]. · Zbl 0712.58059
[56]	Voronov, T.: Graded manifolds and Drinfeld doubles for Lie bialgebroids, In: T. Voronov (ed.), Quantization,Poisson Brackets and Beyond, Contemp. Math. 315, Amer. Math. Soc., Providence, 2002, pp. 131-168. · Zbl 1042.53056
[57]	Wade, A.: Nambu-Dirac structures for Lie algebroids, Lett.Math.Phys. 612002), 85-99. · Zbl 1027.53106