Journal of Lie Theory 28 (2018), No. 3, 735--756
Copyright Heldermann Verlag 2018
Universal Enveloping Algebras and Poincaré-Birkhoff-Witt Theorem for Involutive Hom-Lie Algebras
Li Guo
Dept. of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, P. R. China
and: Dept. of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, U.S.A.
liguo@rutgers.edu
Bin Zhang
School of Mathematics, Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, P. R. China
zhangbin@scu.edu.cn
Shanghua Zheng
Dept. of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, P. R. China
zhengsh@jxnu.edu.cn
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.
Keywords: Hom-Lie algebra, Hom-associative algebra, involution, universal enveloping algebra, Poincaré-Birkhoff-Witt theorem.
MSC: 17A30,17A50,17B35
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