Modules and induced modules of 3-Lie algebra Aω δ
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摘要: 对特征零域
$\mathbb F$ 上无限维单$3$ -李代数$A_{\omega}^{\delta}$ , 构造了两类$A_{\omega}^{\delta}$ 的无限维中间序列模$(V, \rho_{\lambda, 0})=T_{\lambda, 0}$ 与$(V, \rho_{\lambda, 1})=T_{\lambda, 1}$ 和一类无限维ad$(A_{\omega}^{\delta})$ -模$(V, \psi_{\lambda,\mu})$ , 其中$\lambda, \mu\in \mathbb F$ , 并对3-李代数$A_{\omega}^{\delta}$ -模与诱导模之间的关系进行了研究. 证明了只有两类无限维模$(V, \psi_{\lambda,1})$ 和$(V, \psi_{\lambda, 0})$ 是诱导模.Abstract: For the infinite dimensional simple 3-Lie algebra$A_{\omega}^{\delta}$ over a field$\mathbb F$ of characteristic zero, we construct two infinite dimensional intermediate series modules$(V, \rho_{\lambda, 0})=T_{\lambda, 0}$ and$(V, \rho_{\lambda, 1})=T_{\lambda, 1}$ of$A_{\omega}^{\delta}$ as well as a class of infinite dimensional modules$(V, \psi_{\lambda,\mu})$ of ad$(A_{\omega}^{\delta})$ , where$\lambda, \mu\in \mathbb F$ . The relation between 3-Lie algebra$A_{\omega}^{\delta}$ -modules and induced modules of ad$(A_{\omega}^{\delta})$ are discussed. It is shown that only two of infinite dimensional modules, namely$(V, \psi_{\lambda, 1})$ and$(V, \psi_{\lambda, 0})$ , are induced modules.-
Key words:
- 3-Lie algebra-module /
- induced module /
- intermediate series module
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[1] FILIPPOV V. n-Lie algebras [J]. Siberian Mathematical Journal, 1985, 26(6): 126-140. [2] BAI R P, WU Y. Constructions of 3-Lie algebras [J]. Linear Multilinear Algebra, 2015, 63(11): 2171-2186. doi: 10.1080/03081087.2014.986121 [3] AZCARRAGA J A, IZQUIERDO J M. n-ary algebras: A review with applications [J]. Journal of Physics A: Mathematical and Theoretical, 2010, 43(29): 293001. [4] SHENG Y, TANG R. Symplectic, product and complex structures on 3-Lie algebras [J]. Journal of Algebra, 2018, 508: 256-300. doi: 10.1016/j.jalgebra.2018.05.005 [5] BAGGER J, LAMBERT N. Gauge symmetry and supersymmetry of multiple M2-branes [J]. Physical Review D: Particles Fields, 2008, 77(6): 215-240. [6] DEBELLIS J, SAEMANN C, SZABO R J. Quantized Nambu-Poisson manifolds and n-Lie algebras [J]. Journal of Mathematical Physics, 2010, 51(12): 153-306. [7] GUATAVSSON A. Algebraic structures on parallel M2 branes [J]. Nuclear Physics B, 2009, 811(1/2): 66-76. doi: 10.1016/j.nuclphysb.2008.11.014 [8] NAMBU Y. Generalized Hamiltonian Dynamics [J]. Physical Review D: Particles Fields, 1999, 7(8): 2405-2412. [9] TAKHTAJAN L. On foundation of the generalized Nambu mechanics [J]. Communications in Mathematical Physics, 1994, 160(2): 295-315. doi: 10.1007/BF02103278 [10] GAUTHERON P. Some remarks concerning Nambu mechanics [J]. Letters in Mathematical Physics, 1996, 37(1): 103-116. doi: 10.1007/BF00400143 [11] BAI R P, BAI C M, WANG J X. Realizations of 3-Lie algebras [J]. Journal of Mathematical Physics, 2010, 51(6): 063505. doi: 10.1063/1.3436555 [12] BAI R, LI Z H, WANG W D. Infinite-dimensional 3-Lie algebras and their connections to Harish-Chandra modules [J]. Frontiers of Mathematics in China, 2017, 12(3): 515-530. doi: 10.1007/s11464-017-0606-7 [13] KASYMOV S M. Theory of n-Lie algebras [J]. Algebra and Logic, 1987, 26(3): 155-166. doi: 10.1007/BF02009328 [14] LIU J, MAKHLOUF A, SHENG Y. A new approach to representations of 3-Lie algebras and abelian extensions [J]. Algebra Representation Theory, 2017, 20: 1415-1431. doi: 10.1007/s10468-017-9693-0
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