Advanced Search
Article Contents
Article Contents

From Schouten to Mackenzie: Notes on brackets

In Memory of Kirill Mackenzie (1951–2020)

Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 01A60, 17-03, 53-03; Secondary: 17B70, 16E40, 53C15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
  • 加载中

Article Metrics

HTML views(293) PDF downloads(150) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint