{"title":"李氏代数群的模类$ \\infty $与伴随表示","authors":"R. Caseiro, C. Laurent-Gengoux","doi":"10.3934/jgm.2022008","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>We study the modular class of <inline-formula><tex-math id=\"M2\">\\begin{document}$ Q $\\end{document}</tex-math></inline-formula>-manifolds, and in particular of negatively graded Lie <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\infty $\\end{document}</tex-math></inline-formula>-algebroid. We show the equivalence of several descriptions of those classes, that it matches the classes introduced by various authors and that the notion is homotopy invariant. In the process, the adjoint and coadjoint actions up to homotopy of a Lie <inline-formula><tex-math id=\"M4\">\\begin{document}$ \\infty $\\end{document}</tex-math></inline-formula>-algebroid are spelled out. We also wrote down explicitly some dualities, e.g. between representations up to homotopies of Lie <inline-formula><tex-math id=\"M5\">\\begin{document}$ \\infty $\\end{document}</tex-math></inline-formula>-algebroids and their <inline-formula><tex-math id=\"M6\">\\begin{document}$ Q $\\end{document}</tex-math></inline-formula>-manifold equivalent, which we hope to be of use for future reference.</p>","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"17 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Modular class of Lie $ \\\\infty $-algebroids and adjoint representations\",\"authors\":\"R. Caseiro, C. Laurent-Gengoux\",\"doi\":\"10.3934/jgm.2022008\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p style='text-indent:20px;'>We study the modular class of <inline-formula><tex-math id=\\\"M2\\\">\\\\begin{document}$ Q $\\\\end{document}</tex-math></inline-formula>-manifolds, and in particular of negatively graded Lie <inline-formula><tex-math id=\\\"M3\\\">\\\\begin{document}$ \\\\infty $\\\\end{document}</tex-math></inline-formula>-algebroid. We show the equivalence of several descriptions of those classes, that it matches the classes introduced by various authors and that the notion is homotopy invariant. In the process, the adjoint and coadjoint actions up to homotopy of a Lie <inline-formula><tex-math id=\\\"M4\\\">\\\\begin{document}$ \\\\infty $\\\\end{document}</tex-math></inline-formula>-algebroid are spelled out. We also wrote down explicitly some dualities, e.g. between representations up to homotopies of Lie <inline-formula><tex-math id=\\\"M5\\\">\\\\begin{document}$ \\\\infty $\\\\end{document}</tex-math></inline-formula>-algebroids and their <inline-formula><tex-math id=\\\"M6\\\">\\\\begin{document}$ Q $\\\\end{document}</tex-math></inline-formula>-manifold equivalent, which we hope to be of use for future reference.</p>\",\"PeriodicalId\":49161,\"journal\":{\"name\":\"Journal of Geometric Mechanics\",\"volume\":\"17 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2022-03-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Geometric Mechanics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3934/jgm.2022008\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Geometric Mechanics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/jgm.2022008","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 2
摘要
We study the modular class of \begin{document}$ Q $\end{document}-manifolds, and in particular of negatively graded Lie \begin{document}$ \infty $\end{document}-algebroid. We show the equivalence of several descriptions of those classes, that it matches the classes introduced by various authors and that the notion is homotopy invariant. In the process, the adjoint and coadjoint actions up to homotopy of a Lie \begin{document}$ \infty $\end{document}-algebroid are spelled out. We also wrote down explicitly some dualities, e.g. between representations up to homotopies of Lie \begin{document}$ \infty $\end{document}-algebroids and their \begin{document}$ Q $\end{document}-manifold equivalent, which we hope to be of use for future reference.
Modular class of Lie $ \infty $-algebroids and adjoint representations
We study the modular class of \begin{document}$ Q $\end{document}-manifolds, and in particular of negatively graded Lie \begin{document}$ \infty $\end{document}-algebroid. We show the equivalence of several descriptions of those classes, that it matches the classes introduced by various authors and that the notion is homotopy invariant. In the process, the adjoint and coadjoint actions up to homotopy of a Lie \begin{document}$ \infty $\end{document}-algebroid are spelled out. We also wrote down explicitly some dualities, e.g. between representations up to homotopies of Lie \begin{document}$ \infty $\end{document}-algebroids and their \begin{document}$ Q $\end{document}-manifold equivalent, which we hope to be of use for future reference.
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.