September  2021, 13(3): 501-516. doi: 10.3934/jgm.2021014

Brackets by any other name

109 Holly Dr, Lansdale PA 19446, USA

In Memory of Kirill Mackenzie (1951-2020)

Received  March 2021 Published  September 2021 Early access  July 2021

Brackets by another name - Whitehead or Samelson products - have a history parallel to that in Kosmann-Schwarzbach's "From Schouten to Mackenzie: notes on brackets". Here I sketch the development of these and some of the other brackets and products and braces within homotopy theory and homological algebra and with applications to mathematical physics.

In contrast to the brackets of Schouten, Nijenhuis and of Gerstenhaber, which involve a relation to another graded product, in homotopy theory many of the brackets are free standing binary operations. My path takes me through many twists and turns; unless particularized, bracket will be the generic term including product and brace. The path leads beyond binary to multi-linear $ n $-ary operations, either for a single $ n $ or for whole coherent congeries of such assembled into what is known now as an $ \infty $-algebra, such as in homotopy Gerstenhaber algebras. It also leads to more subtle invariants. Along the way, attention will be called to interaction with 'physics'; indeed, it has been a two-way street.

Citation: Jim Stasheff. Brackets by any other name. Journal of Geometric Mechanics, 2021, 13 (3) : 501-516. doi: 10.3934/jgm.2021014
References:
[1]

S. A. Abramyan and T. E. Panov, Higher Whitehead products for moment-angle complexes and substitutions of simplicial complexes, Tr. Mat. Inst. Steklova, 305 (2019), 7-28.  doi: 10.4213/tm3995.

[2]

F. Akman, On some generalizations of Batalin-Vilkovisky algebras, J. Pure Appl. Algebra, 120 (1997), 105-141.  doi: 10.1016/S0022-4049(96)00036-9.

[3]

F. Akman, Multibraces on the Hochschild space, J. Pure Appl. Algebra, 167 (2002), 129-163.  doi: 10.1016/S0022-4049(01)00026-3.

[4]

M. L. Albeggiani, Generalizzione di due teoremi, Rendiconti Del Circol0 Matematic0Di Palermo, Translation: Generalization of two Theorems.

[5] A. AyupovB. Omirov and I. Rakhimov, Leibniz Algebras. Structure and Classification, CRC Press, 2019. 
[6]

H.-J. BauesD. Blanc and S. Gondhali, Higher Toda brackets and Massey products, J. Homotopy Relat. Struct., 11 (2016), 643-677.  doi: 10.1007/s40062-016-0157-8.

[7]

F. Bayen and M. Flato, Remarks concerning Nambu's generalized mechanics, Phys. Rev. D (3), 11 (1975), 3049-3053.  doi: 10.1103/PhysRevD.11.3049.

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F. BayenM. FlatoC. FronsdalA. Lichnerowicz and D. Sternheimer, Deformation theory and quantization. Ⅰ. Deformations of symplectic structures, Ann. Physics, 111 (1978), 61-110.  doi: 10.1016/0003-4916(78)90224-5.

[9]

F. BayenM. FlatoC. FronsdalA. Lichnerowicz and D. Sternheimer, Deformation theory and quantization. Ⅱ. Physical applications, Ann. Physics, 111 (1978), 111-151.  doi: 10.1016/0003-4916(78)90225-7.

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F. BerendsG. Burgers and H. van Dam, On the theoretical problems in constructing intereactions involving higher spin massless particles, Nucl.Phys.B, 260 (1985), 295-322.  doi: 10.1016/0550-3213(85)90074-4.

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A. Bloh, On a generalization of the concept of Lie algebra, Dokl. Akad. Nauk SSSR, 165 (1965), 471-473. 

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R. Bonezzi and O. Hohm, Leibniz gauge theories and infinity structures, Comm. Math. Phys., 377 (2020), 2027-2077.  doi: 10.1007/s00220-020-03785-2.

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J. A. de Azcárraga and J. M. Izquierdo, $n$-ary algebras: A review with applications, J. Phys. A, 43.

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J. A. de Azcárraga and J. C. Pérez Bueno, Higher-order simple Lie algebras, Comm. Math. Phys., 184 (1997), 669-681.  doi: 10.1007/s002200050079.

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P. Deligne, Letter from Deligne to Stasheff, Gerstenhaber, May, Schechtman and Drinfeld, May.

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I. Y. Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A, 125 (1987), 240-246.  doi: 10.1016/0375-9601(87)90201-5.

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A. Douady, Obstruction primaire á la déformation, Séminarie Henri Cartan, Exposé 4.

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F. J. Dyson, Missed opportunities, Bull. Amer. Math. Soc., 78 (1972), 635-652.  doi: 10.1090/S0002-9904-1972-12971-9.

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V. T. Filippov, $n$-ary lie algebras, Sibirskii Math. J., 26 (1985), 126-140. 

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T. Friedmann, P. Hanlon, R. P. Stanley and M. L. Wachs, Action of the symmetric group on the free LAnKe: A CataLAnKe theorem, Sém. Lothar. Combin., 80B (2018), Art. 63, 12pp.

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R. Fulp, T. Lada and J. Stasheff, Sh-Lie algebras induced by gauge transformations, Comm. Math. Phys., 231 (2002), 25–43, arXiv: math.QA/0012106. doi: 10.1007/s00220-002-0678-3.

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C. Gauss, Zur mathematischen theorie der electrodynamischen wirkungen, in Werke, 1877,601–630. doi: 10.1007/978-3-642-49319-5_42.

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M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math., 78 (1963), 267-288.  doi: 10.2307/1970343.

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M. Gerstenhaber and A. A. Voronov, Homotopy $G$-algebras and moduli space operad, Internat. Math. Res. Notices, 1995 (1995), 141–153 (electronic). doi: 10.1155/S1073792895000110.

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E. Getzler, Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology, in Quantum Deformations of Algebras and Their Representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), vol. 7 of Israel Math. Conf. Proc., Bar-Ilan Univ., Ramat Gan, 1993, 65–78.

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V. Gnedbaye, Operads of $k$-ary algebras, in Operads: Proceedings of Renaissance Conferences (eds. J.-L. Loday, J. Stasheff and A. A. Voronov), vol. 202 of Contemporary Mathematics, Amer. Math. Soc., 1997, 83–113. doi: 10.1090/conm/202/02596.

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A. Gracia-SazM. Jotz LeanK. C. H. Mackenzie and R. A. Mehta, Double Lie algebroids and representations up to homotopy, J. Homotopy Relat. Struct., 13 (2018), 287-319.  doi: 10.1007/s40062-017-0183-1.

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P. Hanlon and M. L. Wachs, On Lie $k$-algebras, Adv. in Math., 113 (1995), 206-236.  doi: 10.1006/aima.1995.1038.

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K. A. Hardie, Higher Whitehead products, Quart. J. Math. Oxford Ser. (2), 12 (1961), 241-249.  doi: 10.1093/qmath/12.1.241.

[30]

K. Haring, On the Events Leading to the Formulation of the Gerstenhaber Algebra: 1945-1966, Master's thesis, UNC-CH, 1995.

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P. J. Hilton and J. H. C. Whitehead, Note on the Whitehead product, Ann. of Math. (2), 58 (1953), 429-442.  doi: 10.2307/1969746.

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O. Hohm and B. Zwiebach, $L_{\infty}$ algebras and field theory, Fortsch. Phys., 65 (2017), 1700014..R. 163 doi: 10.1002/prop.201700014.

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J. Huebschmann, On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras, 2021

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T. Kadeishvili, On the homology theory of fibre spaces, Uspekhi Mat. Nauk, 35 (1980), 183–188, arXiv: math.AT/0504437.

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Y. Kosmann-Schwarzbach, From Schouten to Mackenzie: Notes on brackets, 2021.

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Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier (Grenoble), 46 (1996), 1243-1274.  doi: 10.5802/aif.1547.

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Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys., 69 (2004), 61-87.  doi: 10.1007/s11005-004-0608-8.

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Y. Kosmann-Schwarzbach, The Noether Theorems, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, 2011, Invariance and conservation laws in the twentieth century, Translated, revised and augmented from the 2006 French edition by Bertram E. Schwarzbach. doi: 10.1007/978-0-387-87868-3.

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A. Kotov and T. Strobl, The embedding tensor, Leibniz-Loday algebras, and their higher gauge theories, Comm. Math. Phys., 376 (2020), 235-258.  doi: 10.1007/s00220-019-03569-3.

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T. Lada and M. Markl, Strongly homotopy Lie algebras, Comm. in Algebra, 23 (1995), 2147–2161, arXiv: hep-th/9406095. doi: 10.1080/00927879508825335.

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T. Lada and M. Markl, Symmetric brace algebras, Appl. Categ. Structures, 13 (2005), 351-370.  doi: 10.1007/s10485-005-0911-2.

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T. Lada and J. Stasheff, Introduction to sh Lie algebras for physicists, Intern'l J. Theor. Phys., 32 (1993), 1087-1103.  doi: 10.1007/BF00671791.

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S. Lavau and Stasheff.J, From differential crossed modules to tensor hierarchies, arXiv: 2003.07838.

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S. Lavau, Tensor hierarchies and Leibniz algebras, J. Geom. Phys., 144 (2019), 147-189.  doi: 10.1016/j.geomphys.2019.05.014.

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B. H. Lian and G. J. Zuckerman, New perspectives on the BRST-algebraic structure of string theory, Commun. Math. Phys., 154 (1993), 613–646, arXiv: Hep-th/9211072. doi: 10.1007/BF02102111.

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Z.-J. LiuA. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Diff. Geom., 45 (1997), 547-574. 

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J.-L. Loday, Une version non commutative des algebres de Lie: Les algebres de Leibniz, Enseign. Math. (2), 39 (1993), 269-293. 

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A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields, Indag. Math., 17 (1955), 390–397,398–403.

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show all references

References:
[1]

S. A. Abramyan and T. E. Panov, Higher Whitehead products for moment-angle complexes and substitutions of simplicial complexes, Tr. Mat. Inst. Steklova, 305 (2019), 7-28.  doi: 10.4213/tm3995.

[2]

F. Akman, On some generalizations of Batalin-Vilkovisky algebras, J. Pure Appl. Algebra, 120 (1997), 105-141.  doi: 10.1016/S0022-4049(96)00036-9.

[3]

F. Akman, Multibraces on the Hochschild space, J. Pure Appl. Algebra, 167 (2002), 129-163.  doi: 10.1016/S0022-4049(01)00026-3.

[4]

M. L. Albeggiani, Generalizzione di due teoremi, Rendiconti Del Circol0 Matematic0Di Palermo, Translation: Generalization of two Theorems.

[5] A. AyupovB. Omirov and I. Rakhimov, Leibniz Algebras. Structure and Classification, CRC Press, 2019. 
[6]

H.-J. BauesD. Blanc and S. Gondhali, Higher Toda brackets and Massey products, J. Homotopy Relat. Struct., 11 (2016), 643-677.  doi: 10.1007/s40062-016-0157-8.

[7]

F. Bayen and M. Flato, Remarks concerning Nambu's generalized mechanics, Phys. Rev. D (3), 11 (1975), 3049-3053.  doi: 10.1103/PhysRevD.11.3049.

[8]

F. BayenM. FlatoC. FronsdalA. Lichnerowicz and D. Sternheimer, Deformation theory and quantization. Ⅰ. Deformations of symplectic structures, Ann. Physics, 111 (1978), 61-110.  doi: 10.1016/0003-4916(78)90224-5.

[9]

F. BayenM. FlatoC. FronsdalA. Lichnerowicz and D. Sternheimer, Deformation theory and quantization. Ⅱ. Physical applications, Ann. Physics, 111 (1978), 111-151.  doi: 10.1016/0003-4916(78)90225-7.

[10]

F. BerendsG. Burgers and H. van Dam, On the theoretical problems in constructing intereactions involving higher spin massless particles, Nucl.Phys.B, 260 (1985), 295-322.  doi: 10.1016/0550-3213(85)90074-4.

[11]

A. Bloh, On a generalization of the concept of Lie algebra, Dokl. Akad. Nauk SSSR, 165 (1965), 471-473. 

[12]

R. Bonezzi and O. Hohm, Leibniz gauge theories and infinity structures, Comm. Math. Phys., 377 (2020), 2027-2077.  doi: 10.1007/s00220-020-03785-2.

[13]

J. A. de Azcárraga and J. M. Izquierdo, $n$-ary algebras: A review with applications, J. Phys. A, 43.

[14]

J. A. de Azcárraga and J. C. Pérez Bueno, Higher-order simple Lie algebras, Comm. Math. Phys., 184 (1997), 669-681.  doi: 10.1007/s002200050079.

[15]

P. Deligne, Letter from Deligne to Stasheff, Gerstenhaber, May, Schechtman and Drinfeld, May.

[16]

I. Y. Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A, 125 (1987), 240-246.  doi: 10.1016/0375-9601(87)90201-5.

[17]

A. Douady, Obstruction primaire á la déformation, Séminarie Henri Cartan, Exposé 4.

[18]

F. J. Dyson, Missed opportunities, Bull. Amer. Math. Soc., 78 (1972), 635-652.  doi: 10.1090/S0002-9904-1972-12971-9.

[19]

V. T. Filippov, $n$-ary lie algebras, Sibirskii Math. J., 26 (1985), 126-140. 

[20]

T. Friedmann, P. Hanlon, R. P. Stanley and M. L. Wachs, Action of the symmetric group on the free LAnKe: A CataLAnKe theorem, Sém. Lothar. Combin., 80B (2018), Art. 63, 12pp.

[21]

R. Fulp, T. Lada and J. Stasheff, Sh-Lie algebras induced by gauge transformations, Comm. Math. Phys., 231 (2002), 25–43, arXiv: math.QA/0012106. doi: 10.1007/s00220-002-0678-3.

[22]

C. Gauss, Zur mathematischen theorie der electrodynamischen wirkungen, in Werke, 1877,601–630. doi: 10.1007/978-3-642-49319-5_42.

[23]

M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math., 78 (1963), 267-288.  doi: 10.2307/1970343.

[24]

M. Gerstenhaber and A. A. Voronov, Homotopy $G$-algebras and moduli space operad, Internat. Math. Res. Notices, 1995 (1995), 141–153 (electronic). doi: 10.1155/S1073792895000110.

[25]

E. Getzler, Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology, in Quantum Deformations of Algebras and Their Representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), vol. 7 of Israel Math. Conf. Proc., Bar-Ilan Univ., Ramat Gan, 1993, 65–78.

[26]

V. Gnedbaye, Operads of $k$-ary algebras, in Operads: Proceedings of Renaissance Conferences (eds. J.-L. Loday, J. Stasheff and A. A. Voronov), vol. 202 of Contemporary Mathematics, Amer. Math. Soc., 1997, 83–113. doi: 10.1090/conm/202/02596.

[27]

A. Gracia-SazM. Jotz LeanK. C. H. Mackenzie and R. A. Mehta, Double Lie algebroids and representations up to homotopy, J. Homotopy Relat. Struct., 13 (2018), 287-319.  doi: 10.1007/s40062-017-0183-1.

[28]

P. Hanlon and M. L. Wachs, On Lie $k$-algebras, Adv. in Math., 113 (1995), 206-236.  doi: 10.1006/aima.1995.1038.

[29]

K. A. Hardie, Higher Whitehead products, Quart. J. Math. Oxford Ser. (2), 12 (1961), 241-249.  doi: 10.1093/qmath/12.1.241.

[30]

K. Haring, On the Events Leading to the Formulation of the Gerstenhaber Algebra: 1945-1966, Master's thesis, UNC-CH, 1995.

[31]

P. J. Hilton and J. H. C. Whitehead, Note on the Whitehead product, Ann. of Math. (2), 58 (1953), 429-442.  doi: 10.2307/1969746.

[32]

O. Hohm and B. Zwiebach, $L_{\infty}$ algebras and field theory, Fortsch. Phys., 65 (2017), 1700014..R. 163 doi: 10.1002/prop.201700014.

[33]

J. Huebschmann, On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras, 2021

[34]

T. Kadeishvili, On the homology theory of fibre spaces, Uspekhi Mat. Nauk, 35 (1980), 183–188, arXiv: math.AT/0504437.

[35]

Y. Kosmann-Schwarzbach, From Schouten to Mackenzie: Notes on brackets, 2021.

[36]

Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier (Grenoble), 46 (1996), 1243-1274.  doi: 10.5802/aif.1547.

[37]

Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys., 69 (2004), 61-87.  doi: 10.1007/s11005-004-0608-8.

[38]

Y. Kosmann-Schwarzbach, The Noether Theorems, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, 2011, Invariance and conservation laws in the twentieth century, Translated, revised and augmented from the 2006 French edition by Bertram E. Schwarzbach. doi: 10.1007/978-0-387-87868-3.

[39]

A. Kotov and T. Strobl, The embedding tensor, Leibniz-Loday algebras, and their higher gauge theories, Comm. Math. Phys., 376 (2020), 235-258.  doi: 10.1007/s00220-019-03569-3.

[40]

T. Lada and M. Markl, Strongly homotopy Lie algebras, Comm. in Algebra, 23 (1995), 2147–2161, arXiv: hep-th/9406095. doi: 10.1080/00927879508825335.

[41]

T. Lada and M. Markl, Symmetric brace algebras, Appl. Categ. Structures, 13 (2005), 351-370.  doi: 10.1007/s10485-005-0911-2.

[42]

T. Lada and J. Stasheff, Introduction to sh Lie algebras for physicists, Intern'l J. Theor. Phys., 32 (1993), 1087-1103.  doi: 10.1007/BF00671791.

[43]

S. Lavau and Stasheff.J, From differential crossed modules to tensor hierarchies, arXiv: 2003.07838.

[44]

S. Lavau, Tensor hierarchies and Leibniz algebras, J. Geom. Phys., 144 (2019), 147-189.  doi: 10.1016/j.geomphys.2019.05.014.

[45]

B. H. Lian and G. J. Zuckerman, New perspectives on the BRST-algebraic structure of string theory, Commun. Math. Phys., 154 (1993), 613–646, arXiv: Hep-th/9211072. doi: 10.1007/BF02102111.

[46]

Z.-J. LiuA. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Diff. Geom., 45 (1997), 547-574. 

[47]

J.-L. Loday, Une version non commutative des algebres de Lie: Les algebres de Leibniz, Enseign. Math. (2), 39 (1993), 269-293. 

[48]

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

[49]

W. S. Massey, Some higher order cohomology operations, in International Conference on Algebraic Topology, 1958,145–154.

[50]

J. P. May, The Geometry of Iterated Loop Spaces, vol. 271 of Lecture Notes in Math., Springer-Verlag, 1972.

[51]

J. E. McClure and J. H. Smith, A solution of Deligne's Hochschild cohomology conjecture, in Recent Progress in Homotopy Theory (Baltimore, MD, 2000), vol. 293 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2002,153–193. doi: 10.1090/conm/293/04948.

[52]

Y. Nambu, Generalized Hamiltonian dynamics, Phys. Rev. D (3), 7 (1973), 2405-2412.  doi: 10.1103/PhysRevD.7.2405.

[53]

A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields, Indag. Math., 17 (1955), 390–397,398–403.

[54]

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