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.
Categories: math.RA