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《山东大学学报(理学版)》 ›› 2026, Vol. 61 ›› Issue (4): 56-61.doi: 10.6040/j.issn.1671-9352.0.2024.044

• • 上一篇    

3-李代数的导子扩张和Wells正合列

徐森荣1,赵嘉2*   

  1. 1.江苏大学数学科学学院, 江苏 镇江 212013;2.南通大学理学院, 江苏 南通 226019
  • 发布日期:2026-04-08
  • 通讯作者: 赵嘉(1989— ),男,讲师,博士,研究方向为高阶李理论. E-mail:zhaojia@ntu.edu.cn
  • 作者简介:徐森荣(1990— ),男,副教授,博士,研究方向为李代数. E-mail:senrxu@163.com*通信作者:赵嘉(1989— ),男,讲师,博士,研究方向为高阶李理论. E-mail:zhaojia@ntu.edu.cn
  • 基金资助:
    国家自然科学基金资助项目(12201253);江苏省自然科学基金资助项目(BK20220510)

Derivation extensions and Wells exact sequences of 3-Lie algebras

XU Senrong1, ZHAO Jia2*   

  1. 1. School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, Jiangsu, China;
    2. School of Sciences, Nantong University, Nantong 226019, Jiangsu, China
  • Published:2026-04-08

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摘要： 给定一个3-李代数扩张0→AiLpB→0满足［A,A,L］L=0, 其中i:A→L是包含映射。本文建立3-李代数的1-阶闭链、导子对和2-阶上同调群之间联系的Wells正合列,特别地,当上述扩张可裂时, 证明对应的Wells正合列约化为一个短正合列, 并且也是可裂的。

关键词: 3-李代数, 扩张, 导子, 上同调, Wells正合列

Abstract: Given an extension of 3-Lie algebras of the form 0→AiLpB→ 0 with ［A,A,L］L=0, where i:A→L is the inclusion map. In this paper, we establish the Wells exact sequence which relates 1-order cocycles, derivation pairs and the second cohomology groups of 3-Lie algebras. In particular, when the above extension is split, we show that the corresponding Wells exact sequence is reduced to a short exact sequence and is also split.

Key words: 3-Lie algebra, extension, derivation, cohomology, Wells exact sequence

中图分类号: 

  • O152
[1] FILIPPOV V T. n-Lie algebras[J]. Siberian Mathematical Journal, 1985, 26(6):879-891.
[2] NAMBU Y. Generalized Hamiltonian dynamics[J]. Physical Review D, 1973, 7:2405-2412.
[3] CHERKIS S, SÄMANN C. Multiple M2-branes and generalized 3-Lie algebras[J]. Physical Review D, 2008, 78(6):1-11.
[4] DE MEDEIROS P, FIGUEROA-OFARRILL J, MÉNDEZ-ESCOBAR E, et al. On the Lie-algebraic origin of metric 3-algebras[J]. Communications in Mathematical Physics, 2009, 290(3):871-902.
[5] SHENG Yunhe, TANG Rong. Symplectic, product and complex structures on 3-Lie algebras[J]. Journal of Algebra, 2018, 508:256-300.
[6] KASYMOV S M. On a theory of n-Lie algebras[J]. Algebra and logic, 1987, 26(3):155-166.
[7] FIGUEROA-O’FARRILL J. Deformations of 3-algebras[J]. Journal of Mathematical Physics, 2009, 50(11):1-27.
[8] TAKHTAJAN L. Higher order analog of Chevalley-Eilenberg complex and deformation theory of n-gebras[J]. St Petersburg Mathematical Jornal, 1995, 6(2):429-438.
[9] TANG Rong, BAI Chengming, GUO Li, et al. Deformations and their controlling cohomologies of O -operators[J]. Communications in Mathematical Physics, 2019, 368(2):665-700.
[10] 白瑞蒲,刘培. 3-李代数T的齐性Rota-Baxter算子[J]. 山东大学学报(理学版),2021,56(8):61-66. BAI Ruipu, LIU Pei. Homogeneous Rota-Baxter operators of 3-Lie algebra T[J]. Journal of Shandong University(Natural Science), 2021, 56(8):61-66.
[11] 徐森荣,谭易兰,赵嘉. 相对罗巴算子的拟迹函数方法和上同调[J]. 吉林大学学报(理学版),2023,61(6):1313-1318. XU Senrong, TAN Yilan, ZHAO Jia. Quasi-trace function method and cohomology of relative Rota-Baxter operators[J]. Journal of Jilin University(Science Edition), 2023, 61(6):1313-1318.
[12] WELLS C. Automorphisms of group extensions[J]. Transactions of the American Mathematical Society, 1971, 155:189-194.
[13] JIN Ping, LIU Heguo. The Wells exact sequence for the automorphism group of a group extension[J]. Journal of Algebra, 2010, 324(6):1219-1228.
[14] JAMALI A R. The Wells exact sequence for the automorphism group of a Lie ring extension[J]. Journal of Algebra and Its Applications, 2019, 18(3):1-15.
[15] BARDAKOV V G, SINGH M. Extensions and automorphisms of Lie algebras[J]. Journal of Algebra and Its Applications, 2017, 16(9):1-15.
[16] HAZRA S K, HABIB A. Wells exact sequence and automorphisms of extensions of Lie superalgebras[J]. Journal of Lie Theory, 2020, 30(1):179-199.
[17] GOSWAMI S, MISHRA S K, MUKHERJEE G. Automorphisms of extensions of Lie-Yamaguti algebras and inducibility problem[J]. Journal of Algebra, 2024, 641:268-306.
[18] XU Senrong. Cohomology, derivations and abelian extensions of 3-Lie algebras[J]. Journal of Algebra and Its Applications, 2019, 18(7):1-26.
[19] LIU J, MAKHLOUF A, SHENG Y. A new approach to representations of 3-Lie algebras and abelian extensions[J]. Algebra Representation Theory, 2017, 20(6):1415-1431.
[20] ZHANG Tao. Cohomology and deformations of 3-Lie colour algebras[J]. Linear and Multilinear Algebra, 2015, 63(4):651-671.
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