Commutator identities, Lie product identities, and Multiplicative Lie algebras

The aim of this paper is to give a formal meaning of universal commutator identities and to see as to how the structure of multiplicative Lie algebra on a group is a potential candidate to describe universal commutator identities. We introduce some higher Schur multipliers. We shall also introduce and study Steinberg and Chevalley Multiplicative Lie algebras, the multiplicative Lie algebras with commutator identities among the root vectors describing Lie product relations. Free multiplicative Lie algebra on Steinberg and Chevalley groups are discussed.