Derivations, automorphisms, and representations of complex -Lie algebras

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