Heisenberg lie color algebras

In this article, we give a sufficient condition for a Lie color algebra to be complete. The color derivation algebra Der(H) and the holomorph L of finite dimensional Heisenberg Lie color algebra H graded by a torsion-free abelian group over an algebraically closed field of characteristic zero are determined. We prove that Der(H) and Der(L) are simple complete Lie color algebras, but L is not a complete Lie color algebra.