doi: 10.3934/jgm.2021010
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

From Schouten to Mackenzie: Notes on brackets

Paris, France

In Memory of Kirill Mackenzie (1951-2020)

Received  March 2021 Early access May 2021

In this paper, dedicated to the memory of Kirill Mackenzie, I relate the origins and early development of the theory of graded Lie brackets, first in the publications on differential geometry of Schouten, Nijenhuis, and Frölicher–Nijenhuis, then in the work of Gerstenhaber and Nijenhuis–Richardson in cohomology theory.

Citation: Yvette Kosmann-Schwarzbach. From Schouten to Mackenzie: Notes on brackets. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021010
References:
[1]

L. CorwinY. Ne'eman and S. Sternberg, Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry), Rev. Modern Phys., 47 (1975), 573-603.  doi: 10.1103/RevModPhys.47.573.  Google Scholar

[2]

B. Eckmann, Sur les structures complexes et presque complexes, in Géométrie Différentielle (Colloques Internationaux du Cent re National de la Recherche Scientifique, Strasbourg, 1953), Centre National de la Recherche Scientifique, Paris, 1953, 151–159.  Google Scholar

[3]

B. Eckmann and A. Frölicher, Sur l'intégrabilité des structures presque complexes, C. R. Acad. Sci. Paris, 232 (1951), 2284-2286.   Google Scholar

[4]

M. FlatoA. Lichnerowicz and D. Sternheimer, Deformations of Poisson brackets, Dirac brackets and applications, J. Math. Phys., 17 (1976), 1754-1762.  doi: 10.1063/1.523104.  Google Scholar

[5]

A. Frölicher, Zur differentialgeometrie der komplexen strukturen, Math. Ann., 129 (1955), 50-95.  doi: 10.1007/BF01362360.  Google Scholar

[6]

A. Frölicher and A. Nijenhuis, Theory of vector-valued differential forms. I. Derivations of the graded ring of differential forms, Indag. Math., 18 (1956), 338-359.   Google Scholar

[7]

M. Gerstenhaber, The cohomology structure of an associative ring, Annals of Mathematics, Second Series, 78 (1963), 267–288. doi: 10.2307/1970343.  Google Scholar

[8]

M. Gerstenhaber, On the deformation of rings and algebras, Annals of Mathematics, Second Series, 79 (1964), 59–103. doi: 10.2307/1970484.  Google Scholar

[9]

M. Gerstenhaber and S. D. Schack, Algebraic cohomology and deformation theory, in Deformation Theory of Algebras and Structures and Applications, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 247, M. Hazewinkel and M. Gerstenhaber, eds., Kluwer Acad. Publ., Dordrecht, 1988, 11–264. doi: 10.1007/978-94-009-3057-5_2.  Google Scholar

[10]

D. K. Harrison, Commutative algebras and cohomology, Trans. Amer. Math. Soc., 104 (1962), 191-204.  doi: 10.1090/S0002-9947-1962-0142607-6.  Google Scholar

[11]

G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math., 46 (1945), 58-67.  doi: 10.2307/1969145.  Google Scholar

[12]

P. Libermann, Sur le problème d'équivalence de certaines structures infinitésimales, Ann. Mat. Pura Appl., 36 (1954), 27-120.  doi: 10.1007/BF02412833.  Google Scholar

[13]

P. Libermann, Sur les structures presque complexes et autres structures infinitésimales régulières, Bull. Soc. Math. France, 83 (1955), 195-224.  doi: 10.24033/bsmf.1460.  Google Scholar

[14]

K. A. Mackenzie, Rigid cohomology of topological groupoids, J. Austral. Math. Soc., Ser. A. 26 (1978), 277–301. doi: 10.1017/S1446788700011794.  Google Scholar

[15] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathematical Society Lecture Note Series, 124, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511661839.  Google Scholar
[16]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147.  doi: 10.1112/blms/27.2.97.  Google Scholar

[17] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883.  Google Scholar
[18]

A. Nijenhuis, $X_{n-1}$-forming sets of eigenvectors, Indag. Math., 13 (1951), 200-212.   Google Scholar

[19]

A. Nijenhuis, Theory of the Geometric Object, Thesis, University of Amsterdam, Amsterdam, 1952.  Google Scholar

[20]

A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields. I, II, Nederl. Akad. Wetensch. Proc., Ser. A. 58 = Indag. Math., 17 (1955), 390–397, 398–403.  Google Scholar

[21]

A. Nijenhuis, J. A. Schouten: A master at tensors (28 August 1883–20 January 1971), Nieuwe Arch. Wisk., 20 (1972), 1-19.   Google Scholar

[22]

A. Nijenhuis and R. W. Richardson, Cohomology and deformations of algebraic structures, Bull. Amer. Math. Soc., 70 (1964), 406-411.  doi: 10.1090/S0002-9904-1964-11117-4.  Google Scholar

[23]

A. Nijenhuis and R. W. Richardson, Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc., 72 (1966), 1-29.  doi: 10.1090/S0002-9904-1966-11401-5.  Google Scholar

[24]

A. Nijenhuis and R. W. Richardson, Deformations of Lie algebra structures, J. Math. Mech., 17 (1967), 89-105.  doi: 10.1512/iumj.1968.17.17005.  Google Scholar

[25]

E. Noether, Invarianten beliebiger Differentialausdrücke, Gött. Nachr., (1918), 37–44. doi: 10.1007/978-3-642-39990-9_12.  Google Scholar

[26]

J. A. Schouten, Grundlagen der Vektor- und Affinoranalysis, Leipzig, B. G. Teubner, 1914. Google Scholar

[27]

J. A. Schouten, Ueber Differentialkomitanten zweier kontravarianter Grössen, Nederl. Akad. Wetensch. Proc., 43 (1940), 449–452.  Google Scholar

[28]

J. A. Schouten, Sur les tenseurs de $V^{n}$ aux directions principales $V^{n-1}$-normales, in Colloque Géom. diff., Louvain 1951, Georges Thone, Liège; Masson & Cie, Paris, (1951), 67–70.  Google Scholar

[29]

J. A. Schouten, Ricci-Calculus: An Introduction to Tensor Analysis and its Geometrical Applications, Springer, 1954.  Google Scholar

[30]

J. A. Schouten, On the differential operators of first order in tensor calculus, in Convegno Internaz. Geometria Differenz., Italia, 20–26 Settembre 1953, Roma, Edizioni Cremonese (1954), 1–7 (Report 1953-012, Mathematische Centrum, Amsterdam).  Google Scholar

[31]

J. A. Schouten and D. J. Struik, Einführung in die neueren Methoden der Differentialgeometrie (vol. 1, Algebra und Übertragungslehre, von J. A. Schouten; vol. 2, Geometrie, von D. J. Struik), Groningen-Batavia, P. Noordhoff N. V., 1935–1938. Google Scholar

[32]

J. A. Schouten and K. Yano, On the geometrical meaning of the vanishing of the Nijenhuis tensor in an $X_2n$ with an almost complex structure, Nederl. Akad. Wetensch. Proc., Ser. A. 58 = Indag. Math., 17 (1955), 133–138.  Google Scholar

[33]

W. Ślebodziński, Sur les équations de Hamilton, Bull. Acad. Royale de Belgique, 17 (1931), 864-870.   Google Scholar

[34]

D. J. J. A. Struik, Schouten and the tensor calculus, Nieuwe Arch. Wisk., 26 (1978), 96-107.   Google Scholar

[35]

A. Tonolo, Sopra una classe di deformazioni finite, Ann. Mat. Pura Appl., IV. Ser., 29 (1949), 99–114. doi: 10.1007/BF02413917.  Google Scholar

show all references

References:
[1]

L. CorwinY. Ne'eman and S. Sternberg, Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry), Rev. Modern Phys., 47 (1975), 573-603.  doi: 10.1103/RevModPhys.47.573.  Google Scholar

[2]

B. Eckmann, Sur les structures complexes et presque complexes, in Géométrie Différentielle (Colloques Internationaux du Cent re National de la Recherche Scientifique, Strasbourg, 1953), Centre National de la Recherche Scientifique, Paris, 1953, 151–159.  Google Scholar

[3]

B. Eckmann and A. Frölicher, Sur l'intégrabilité des structures presque complexes, C. R. Acad. Sci. Paris, 232 (1951), 2284-2286.   Google Scholar

[4]

M. FlatoA. Lichnerowicz and D. Sternheimer, Deformations of Poisson brackets, Dirac brackets and applications, J. Math. Phys., 17 (1976), 1754-1762.  doi: 10.1063/1.523104.  Google Scholar

[5]

A. Frölicher, Zur differentialgeometrie der komplexen strukturen, Math. Ann., 129 (1955), 50-95.  doi: 10.1007/BF01362360.  Google Scholar

[6]

A. Frölicher and A. Nijenhuis, Theory of vector-valued differential forms. I. Derivations of the graded ring of differential forms, Indag. Math., 18 (1956), 338-359.   Google Scholar

[7]

M. Gerstenhaber, The cohomology structure of an associative ring, Annals of Mathematics, Second Series, 78 (1963), 267–288. doi: 10.2307/1970343.  Google Scholar

[8]

M. Gerstenhaber, On the deformation of rings and algebras, Annals of Mathematics, Second Series, 79 (1964), 59–103. doi: 10.2307/1970484.  Google Scholar

[9]

M. Gerstenhaber and S. D. Schack, Algebraic cohomology and deformation theory, in Deformation Theory of Algebras and Structures and Applications, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 247, M. Hazewinkel and M. Gerstenhaber, eds., Kluwer Acad. Publ., Dordrecht, 1988, 11–264. doi: 10.1007/978-94-009-3057-5_2.  Google Scholar

[10]

D. K. Harrison, Commutative algebras and cohomology, Trans. Amer. Math. Soc., 104 (1962), 191-204.  doi: 10.1090/S0002-9947-1962-0142607-6.  Google Scholar

[11]

G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math., 46 (1945), 58-67.  doi: 10.2307/1969145.  Google Scholar

[12]

P. Libermann, Sur le problème d'équivalence de certaines structures infinitésimales, Ann. Mat. Pura Appl., 36 (1954), 27-120.  doi: 10.1007/BF02412833.  Google Scholar

[13]

P. Libermann, Sur les structures presque complexes et autres structures infinitésimales régulières, Bull. Soc. Math. France, 83 (1955), 195-224.  doi: 10.24033/bsmf.1460.  Google Scholar

[14]

K. A. Mackenzie, Rigid cohomology of topological groupoids, J. Austral. Math. Soc., Ser. A. 26 (1978), 277–301. doi: 10.1017/S1446788700011794.  Google Scholar

[15] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathematical Society Lecture Note Series, 124, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511661839.  Google Scholar
[16]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147.  doi: 10.1112/blms/27.2.97.  Google Scholar

[17] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883.  Google Scholar
[18]

A. Nijenhuis, $X_{n-1}$-forming sets of eigenvectors, Indag. Math., 13 (1951), 200-212.   Google Scholar

[19]

A. Nijenhuis, Theory of the Geometric Object, Thesis, University of Amsterdam, Amsterdam, 1952.  Google Scholar

[20]

A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields. I, II, Nederl. Akad. Wetensch. Proc., Ser. A. 58 = Indag. Math., 17 (1955), 390–397, 398–403.  Google Scholar

[21]

A. Nijenhuis, J. A. Schouten: A master at tensors (28 August 1883–20 January 1971), Nieuwe Arch. Wisk., 20 (1972), 1-19.   Google Scholar

[22]

A. Nijenhuis and R. W. Richardson, Cohomology and deformations of algebraic structures, Bull. Amer. Math. Soc., 70 (1964), 406-411.  doi: 10.1090/S0002-9904-1964-11117-4.  Google Scholar

[23]

A. Nijenhuis and R. W. Richardson, Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc., 72 (1966), 1-29.  doi: 10.1090/S0002-9904-1966-11401-5.  Google Scholar

[24]

A. Nijenhuis and R. W. Richardson, Deformations of Lie algebra structures, J. Math. Mech., 17 (1967), 89-105.  doi: 10.1512/iumj.1968.17.17005.  Google Scholar

[25]

E. Noether, Invarianten beliebiger Differentialausdrücke, Gött. Nachr., (1918), 37–44. doi: 10.1007/978-3-642-39990-9_12.  Google Scholar

[26]

J. A. Schouten, Grundlagen der Vektor- und Affinoranalysis, Leipzig, B. G. Teubner, 1914. Google Scholar

[27]

J. A. Schouten, Ueber Differentialkomitanten zweier kontravarianter Grössen, Nederl. Akad. Wetensch. Proc., 43 (1940), 449–452.  Google Scholar

[28]

J. A. Schouten, Sur les tenseurs de $V^{n}$ aux directions principales $V^{n-1}$-normales, in Colloque Géom. diff., Louvain 1951, Georges Thone, Liège; Masson & Cie, Paris, (1951), 67–70.  Google Scholar

[29]

J. A. Schouten, Ricci-Calculus: An Introduction to Tensor Analysis and its Geometrical Applications, Springer, 1954.  Google Scholar

[30]

J. A. Schouten, On the differential operators of first order in tensor calculus, in Convegno Internaz. Geometria Differenz., Italia, 20–26 Settembre 1953, Roma, Edizioni Cremonese (1954), 1–7 (Report 1953-012, Mathematische Centrum, Amsterdam).  Google Scholar

[31]

J. A. Schouten and D. J. Struik, Einführung in die neueren Methoden der Differentialgeometrie (vol. 1, Algebra und Übertragungslehre, von J. A. Schouten; vol. 2, Geometrie, von D. J. Struik), Groningen-Batavia, P. Noordhoff N. V., 1935–1938. Google Scholar

[32]

J. A. Schouten and K. Yano, On the geometrical meaning of the vanishing of the Nijenhuis tensor in an $X_2n$ with an almost complex structure, Nederl. Akad. Wetensch. Proc., Ser. A. 58 = Indag. Math., 17 (1955), 133–138.  Google Scholar

[33]

W. Ślebodziński, Sur les équations de Hamilton, Bull. Acad. Royale de Belgique, 17 (1931), 864-870.   Google Scholar

[34]

D. J. J. A. Struik, Schouten and the tensor calculus, Nieuwe Arch. Wisk., 26 (1978), 96-107.   Google Scholar

[35]

A. Tonolo, Sopra una classe di deformazioni finite, Ann. Mat. Pura Appl., IV. Ser., 29 (1949), 99–114. doi: 10.1007/BF02413917.  Google Scholar

[1]

Hassan Boualem, Robert Brouzet. Semi-simple generalized Nijenhuis operators. Journal of Geometric Mechanics, 2012, 4 (4) : 385-395. doi: 10.3934/jgm.2012.4.385

[2]

Sobhan Seyfaddini. Spectral killers and Poisson bracket invariants. Journal of Modern Dynamics, 2015, 9: 51-66. doi: 10.3934/jmd.2015.9.51

[3]

Pengliang Xu, Xiaomin Tang. Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra. Electronic Research Archive, 2021, 29 (4) : 2771-2789. doi: 10.3934/era.2021013

[4]

Frol Zapolsky. Quasi-states and the Poisson bracket on surfaces. Journal of Modern Dynamics, 2007, 1 (3) : 465-475. doi: 10.3934/jmd.2007.1.465

[5]

Karina Samvelyan, Frol Zapolsky. Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces. Electronic Research Announcements, 2017, 24: 28-37. doi: 10.3934/era.2017.24.004

[6]

Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017

[7]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1115-1129. doi: 10.3934/dcdss.2020066

[8]

Ermal Feleqi, Franco Rampazzo. Integral representations for bracket-generating multi-flows. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4345-4366. doi: 10.3934/dcds.2015.35.4345

[9]

Francisco Crespo, Francisco Javier Molero, Sebastián Ferrer. Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier. Journal of Geometric Mechanics, 2016, 8 (2) : 169-178. doi: 10.3934/jgm.2016002

[10]

Nancy López Reyes, Luis E. Benítez Babilonia. A discrete hierarchy of double bracket equations and a class of negative power series. Mathematical Control & Related Fields, 2017, 7 (1) : 41-52. doi: 10.3934/mcrf.2017003

[11]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[12]

Yvette Kosmann-Schwarzbach. From Schouten to Mackenzie: Notes on brackets. Journal of Geometric Mechanics, 2021, 13 (3) : 459-476. doi: 10.3934/jgm.2021013

[13]

Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239

[14]

Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453

[15]

Navin Keswani. Homotopy invariance of relative eta-invariants and $C^*$-algebra $K$-theory. Electronic Research Announcements, 1998, 4: 18-26.

[16]

Theodore Voronov. Book review: General theory of Lie groupoids and Lie algebroids, by Kirill C. H. Mackenzie. Journal of Geometric Mechanics, 2021, 13 (3) : 277-283. doi: 10.3934/jgm.2021026

[17]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, 2021, 29 (3) : 2445-2456. doi: 10.3934/era.2020123

[18]

Neşet Deniz Turgay. On the mod p Steenrod algebra and the Leibniz-Hopf algebra. Electronic Research Archive, 2020, 28 (2) : 951-959. doi: 10.3934/era.2020050

[19]

Hari Bercovici, Viorel Niţică. A Banach algebra version of the Livsic theorem. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 523-534. doi: 10.3934/dcds.1998.4.523

[20]

Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420

2020 Impact Factor: 0.857

Metrics

  • PDF downloads (80)
  • HTML views (182)
  • Cited by (0)

Other articles
by authors

[Back to Top]