2023/10/18 by Luís Mendonça, Mendonça, Luis · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2310.12331
openalex publication_date 2023/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ∗ 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 Gröbner-Shirshov bases.