Some applications of Gr\"obner-Shirshov bases to Lie algebras

Abstract

We show that if a countably generated Lie algebra $H$ does not contain isomorphic copies of certain finite-dimensional nilpotent Lie algebras $A$ and $B$ (satisfying some mild conditions), then $H$ embeds into a quotient of $A \ast B$ that is at the same time hopfian and cohopfian. This is a Lie algebraic version of an embedding theorem proved by C. Miller and P. Schupp for groups. We also prove that any finitely presentable Lie algebra is the quotient of a finitely presented, centerless, residually nilpotent and SQ-universal Lie algebra of cohomological dimension at most $2$ by an ideal that can be generated by two elements as a Lie subalgebra. This is reminiscent of the Rips construction in group theory. In both results we use the theory of Gr\"obner-Shirshov bases.