{"title":"任何其他名称的括号","authors":"J. Stasheff","doi":"10.3934/jgm.2021014","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>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 <i>sketch</i> 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.</p> <p style='text-indent:20px;'>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, <i>bracket</i> will be the generic term including product and brace. The path leads beyond binary to multi-linear <inline-formula><tex-math id=\"M1\">\\begin{document}$ n $\\end{document}</tex-math></inline-formula>-ary operations, either for a single <inline-formula><tex-math id=\"M2\">\\begin{document}$ n $\\end{document}</tex-math></inline-formula> or for whole coherent congeries of such assembled into what is known now as an <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\infty $\\end{document}</tex-math></inline-formula>-algebra, such as in homotopy Gerstenhaber algebras. It also leads to more subtle invariants. Along the way, attention will be called to interaction with 'physics'; indeed, it has been a two-way street.</p>","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"6 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Brackets by any other name\",\"authors\":\"J. Stasheff\",\"doi\":\"10.3934/jgm.2021014\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p style='text-indent:20px;'>Brackets by another name - Whitehead or Samelson products - have a history parallel to that in Kosmann-Schwarzbach's \\\"From Schouten to Mackenzie: notes on brackets\\\". Here I <i>sketch</i> the development of these and some of the other brackets and products and braces within homotopy theory and homological algebra and with applications to mathematical physics.</p> <p style='text-indent:20px;'>In contrast to the brackets of Schouten, Nijenhuis and of Gerstenhaber, which involve a relation to another graded product, in homotopy theory many of the brackets are free standing binary operations. My path takes me through many twists and turns; unless particularized, <i>bracket</i> will be the generic term including product and brace. The path leads beyond binary to multi-linear <inline-formula><tex-math id=\\\"M1\\\">\\\\begin{document}$ n $\\\\end{document}</tex-math></inline-formula>-ary operations, either for a single <inline-formula><tex-math id=\\\"M2\\\">\\\\begin{document}$ n $\\\\end{document}</tex-math></inline-formula> or for whole coherent congeries of such assembled into what is known now as an <inline-formula><tex-math id=\\\"M3\\\">\\\\begin{document}$ \\\\infty $\\\\end{document}</tex-math></inline-formula>-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.</p>\",\"PeriodicalId\":49161,\"journal\":{\"name\":\"Journal of Geometric Mechanics\",\"volume\":\"6 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2021-05-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Geometric Mechanics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3934/jgm.2021014\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Geometric Mechanics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/jgm.2021014","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
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 \begin{document}$ n $\end{document}-ary operations, either for a single \begin{document}$ n $\end{document} or for whole coherent congeries of such assembled into what is known now as an \begin{document}$ \infty $\end{document}-algebra, such as in homotopy Gerstenhaber algebras. It also leads to more subtle invariants. Along the way, attention will be called to interaction with 'physics'; indeed, it has been a two-way street.
Brackets by another name - Whitehead or Samelson products - have a history parallel to that in Kosmann-Schwarzbach's "From Schouten to Mackenzie: notes on brackets". Here I sketch the development of these and some of the other brackets and products and braces within homotopy theory and homological algebra and with applications to mathematical physics.
In contrast to the brackets of Schouten, Nijenhuis and of Gerstenhaber, which involve a relation to another graded product, in homotopy theory many of the brackets are free standing binary operations. My path takes me through many twists and turns; unless particularized, bracket will be the generic term including product and brace. The path leads beyond binary to multi-linear \begin{document}$ n $\end{document}-ary operations, either for a single \begin{document}$ n $\end{document} or for whole coherent congeries of such assembled into what is known now as an \begin{document}$ \infty $\end{document}-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.
期刊介绍:
The Journal of Geometric Mechanics (JGM) aims to publish research articles devoted to geometric methods (in a broad sense) in mechanics and control theory, and intends to facilitate interaction between theory and applications. Advances in the following topics are welcomed by the journal:
1. Lagrangian and Hamiltonian mechanics
2. Symplectic and Poisson geometry and their applications to mechanics
3. Geometric and optimal control theory
4. Geometric and variational integration
5. Geometry of stochastic systems
6. Geometric methods in dynamical systems
7. Continuum mechanics
8. Classical field theory
9. Fluid mechanics
10. Infinite-dimensional dynamical systems
11. Quantum mechanics and quantum information theory
12. Applications in physics, technology, engineering and the biological sciences.