Finite Gröbner–Shirshov bases and faithful representations for Lee monoids

It is shown that every Lee algebra of finite rank admits a finite Gröbner–Shirshov basis, which is proved by constructing a finite complete rewriting system for the corresponding Lee monoid [Formula: see text] for each [Formula: see text]. Further, we explicitly construct a faithful representation for each [Formula: see text], and this faithful representation can be extended to the involution case.