Derivations, Automorphisms, and Representations of Complex $\omega$-Lie Algebras

Abstract

Let $(\mathfrak{g},\omega)$ be a finite-dimensional non-Lie complex $\omega$-Lie algebra. We study the derivation algebra $Der(\mathfrak{g})$ and the automorphism group $Aut(\mathfrak{g})$ of $(\mathfrak{g},\omega)$. We introduce the notions of $\omega$-derivations and $\omega$-automorphisms of $(\mathfrak{g},\omega)$ which naturally preserve the bilinear form $\omega$. We show that the set $Der_{\omega}(\mathfrak{g})$ of all $\omega$-derivations is a Lie subalgebra of $Der(\mathfrak{g})$ and the set $Aut_{\omega}(\mathfrak{g})$ of all $\omega$-automorphisms is a subgroup of $Aut(\mathfrak{g})$. For any 3-dimensional and 4-dimensional nontrivial $\omega$-Lie algebra $\mathfrak{g}$, we compute $Der(\mathfrak{g})$ and $Aut(\mathfrak{g})$ explicitly, and study some Lie group properties of $Aut(\mathfrak{g})$. We also study representation theory of $\omega$-Lie algebras. We show that all 3-dimensional nontrivial $\omega$-Lie algebras are multiplicative, as well as we provide a 4-dimensional example of $\omega$-Lie algebra that is not multiplicative. Finally, we show that any irreducible representation of the simple $\omega$-Lie algebra $C_{\alpha}(\alpha\neq 0,-1)$ is 1-dimensional.