On Low-Dimensional Complex ω-Lie Superalgebras

• Published in: Advances in applied Clifford algebras Vol. 31; no. 3
• Main Authors: Zhou, Jia, Chen, Liangyun
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.07.2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Automorphisms; Mathematical and Computational Physics; Mathematical Methods in Physics; Physics; Physics and Astronomy; Subgroups; Theoretical; 17B60; Derivations; Representations; 17A30; Automorphisms; Lie superalgebra
• ISSN: 0188-7009; 1661-4909
• DOI: 10.1007/s00006-021-01141-8

Let ( g , [ - , - ] , ω ) be a finite-dimensional complex ω -Lie superalgebra. In this paper, we introduce the notions of derivation superalgebra Der ( g ) and the automorphism group Aut ( g ) of ( g , [ - , - ] , ω ) . We study Der ω ( g ) and Aut ω ( g ) , which are superalgebra of Der ( g ) and subgroup of Aut ( g ) , respectively. For any 3-dimensional or 4-dimensional complex ω -Lie superalgebra g , we explicitly calculate Der ( g ) and Aut ( g ) , and obtain Jordan standard forms of elements in the two sets. We also study representation theory of ω -Lie superalgebras and give a conclusion that all nontrivial non- ω -Lie 3-dimensional and 4-dimensional ω -Lie superalgebras are multiplicative, as well as we show that any irreducible respresentation of the 4-dimensional ω -Lie superalgebra P 2 , k ( k ≠ 0 , - 1 ) is 1-dimensional.