Journal of Lie Theory

no. 3
Universal Enveloping Algebras and Poincaré-Birkhoff-Witt Theorem for Involutive Hom-Lie Algebras
Li Guo1, 2; Bin Zhang3; Shanghua Zheng1
1Dept. of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, P. R. China
2Dept. of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, U.S.A.
3School of Mathematics, Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, P. R. China
Journal of Lie Theory, Volume 28 (2018) no. 3, pp. 735-756
Hom-type algebras, in particular Hom-Lie algebras, have attracted quite much attention in recent years. A Hom-Lie algebra is called involutive if its Hom map is multiplicative and involutive. In this paper, we obtain an explicit construction of the free involutive Hom-associative algebra on a Hom-module. We then apply this construction to obtain the universal enveloping algebra of an involutive Hom-Lie algebra. Finally we generalize the well-known Poincaré-Birkhoff-Witt theorem for enveloping algebras of Lie algebras to involutive Hom-Lie algebras.
DOI: 10.5802/jolt.1023
Classification: 17A30,17A50,17B35
Keywords: Hom-Lie algebra, Hom-associative algebra, involution, universal enveloping algebra, Poincaré-Birkhoff-Witt theorem

Li Guo  1 , 2 ; Bin Zhang  3 ; Shanghua Zheng  1

1 Dept. of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, P. R. China
2 Dept. of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, U.S.A.
3 School of Mathematics, Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, P. R. China 
Li Guo; Bin Zhang; Shanghua Zheng. Universal Enveloping Algebras and Poincaré-Birkhoff-Witt Theorem for Involutive Hom-Lie Algebras. Journal of Lie Theory, Volume 28 (2018) no. 3, pp. 735-756. doi: 10.5802/jolt.1023
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     author = {Li Guo and Bin Zhang and Shanghua Zheng},
     title = {Universal {Enveloping} {Algebras} and {Poincar\'e-Birkhoff-Witt} {Theorem} for {Involutive} {Hom-Lie} {Algebras}},
     journal = {Journal of Lie Theory},
     pages = {735--756},
     year = {2018},
     volume = {28},
     number = {3},
     doi = {10.5802/jolt.1023},
     zbl = {1422.17001},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1023/}
}
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