Derivations, automorphisms, and representations of complex ω-Lie algebras. Issue 2 (1st February 2018)

• Record Type: Journal Article
• Authors: Chen, Yin; Zhang, Ziping; Zhang, Runxuan; Zhuang, Rushu
• Is Part Of: Communications in algebra. Volume 46:Issue 2(2018)
• Page Start: 708
• Page End: 726
• Publication Date: 2018-02-01
• Subjects: ω-Lie algebras -- automorphisms -- derivations -- irreducible representations; 17B60 -- 17A30; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2017.1327062
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: Let ( 𝔤, ω ) be a finite-dimensional non-Lie complex ω -Lie algebra. We study the derivation algebra Der ( 𝔤 ) and the automorphism group Aut ( 𝔤 ) of ( 𝔤, ω ). We introduce the notions of ω -derivations and ω -automorphisms of ( 𝔤, ω ) which naturally preserve the bilinear form ω . We show that the set Der ω ( 𝔤 ) of all ω -derivations is a Lie subalgebra of Der ( 𝔤 ) and the set Aut ω ( 𝔤 ) of all ω -automorphisms is a subgroup of Aut ( 𝔤 ). For any three-dimensional and four-dimensional nontrivial ω -Lie algebra 𝔤, we compute Der ( 𝔤 ) and Aut ( 𝔤 ) explicitly, and study some Lie group properties of Aut ( 𝔤 ). We also study representation theory of ω -Lie algebras. We show that all three-dimensional nontrivial ω -Lie algebras are multiplicative, as well as we provide a four-dimensional example of ω -Lie algebra that is not multiplicative. Finally, we show that any irreducible representation of the simple ω -Lie algebra C α ( α ≠0, −1) is one-dimensional.