
Journal of Lie Theory 28 (2018), No. 3, 735756 Copyright Heldermann Verlag 2018 Universal Enveloping Algebras and PoincaréBirkhoffWitt Theorem for Involutive HomLie 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 Homtype algebras, in particular HomLie algebras, have attracted quite much attention in recent years. A HomLie algebra is called involutive if its Hom map is multiplicative and involutive. In this paper, we obtain an explicit construction of the free involutive Homassociative algebra on a Hommodule. We then apply this construction to obtain the universal enveloping algebra of an involutive HomLie algebra. Finally we generalize the wellknown PoincaréBirkhoffWitt theorem for enveloping algebras of Lie algebras to involutive HomLie algebras. Keywords: HomLie algebra, Homassociative algebra, involution, universal enveloping algebra, PoincaréBirkhoffWitt theorem. MSC: 17A30,17A50,17B35 [ Fulltextpdf (156 KB)] for subscribers only. 