September  2021, 13(3): 459-476. doi: 10.3934/jgm.2021013

From Schouten to Mackenzie: Notes on brackets

Paris, France

In Memory of Kirill Mackenzie (1951–2020)

Received  March 2021 Published  September 2021 Early access  July 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, 2021, 13 (3) : 459-476. doi: 10.3934/jgm.2021013
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.

[2]

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

[3]

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

[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.

[5]

A. Frölicher, Zur Differentialgeometrie der komplexen Strukturen, Math. Ann., 129 (1955), 50-95.  doi: 10.1007/BF01362360.

[6]

A. Frölicher and A. Nijenhuis, Theory of vector-valued differential forms. I. Derivations of the graded ring of differential forms, Nederl. Akad. Wetensch. Proc. Ser. A., 59 = Indag. Math., 18 (1956), 338-359. 

[7]

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

[8]

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

[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.

[10]

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

[11]

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

[12]

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

[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.

[14]

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

[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.

[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.

[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.

[18]

A. Nijenhuis, $X_{n-1}$-forming sets of eigenvectors, Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indag. Math., 13 (1951), 200-212. 

[19]

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

[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.

[21]

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

[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.

[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.

[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.

[25]

E. Noether, Invarianten beliebiger Differentialausdrücke, Gött. Nachr. (1918), 37–44.

[26]

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

[27]

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

[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.

[29]

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

[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).

[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.

[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. 

[33]

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

[34]

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

[35]

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

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.

[2]

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

[3]

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

[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.

[5]

A. Frölicher, Zur Differentialgeometrie der komplexen Strukturen, Math. Ann., 129 (1955), 50-95.  doi: 10.1007/BF01362360.

[6]

A. Frölicher and A. Nijenhuis, Theory of vector-valued differential forms. I. Derivations of the graded ring of differential forms, Nederl. Akad. Wetensch. Proc. Ser. A., 59 = Indag. Math., 18 (1956), 338-359. 

[7]

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

[8]

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

[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.

[10]

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

[11]

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

[12]

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

[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.

[14]

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

[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.

[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.

[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.

[18]

A. Nijenhuis, $X_{n-1}$-forming sets of eigenvectors, Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indag. Math., 13 (1951), 200-212. 

[19]

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

[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.

[21]

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

[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.

[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.

[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.

[25]

E. Noether, Invarianten beliebiger Differentialausdrücke, Gött. Nachr. (1918), 37–44.

[26]

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

[27]

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

[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.

[29]

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

[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).

[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.

[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. 

[33]

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

[34]

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

[35]

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

[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]

Christopher Griffin, James Fan. Control problems with vanishing Lie Bracket arising from complete odd circulant evolutionary games. Journal of Dynamics and Games, 2022, 9 (2) : 165-189. doi: 10.3934/jdg.2022002

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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 and 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]

Editorial Office. Retraction: From Schouten to Mackenzie: Notes on brackets. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021010

[17]

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

[18]

Mark Wilkinson. A Lie algebra-theoretic approach to characterisation of collision invariants of the Boltzmann equation for general convex particles. Kinetic and Related Models, 2022, 15 (2) : 283-315. doi: 10.3934/krm.2022008

[19]

Paul Breiding, Türkü Özlüm Çelik, Timothy Duff, Alexander Heaton, Aida Maraj, Anna-Laura Sattelberger, Lorenzo Venturello, Oǧuzhan Yürük. Nonlinear algebra and applications. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021045

[20]

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

2020 Impact Factor: 0.857

Metrics

  • PDF downloads (113)
  • HTML views (232)
  • Cited by (0)

Other articles
by authors

[Back to Top]