Growth of subalgebras and subideals in free Lie algebras

2013/07/04 by Yuri Bahturin, Bahturin, Yuri, Alexander Olshanskii +1 · 1 citation
Mathematics · #16P90 #17B01 #17B40 #17B65 #Advanced Topics in Algebra #Advanced Topology and Set Theory #FOS: Mathematics #Meromorphic and Entire Functions #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1307.1438

openalex publication_date 2013/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate subalgebras in free Lie algebras, the main tool being relative growth and cogrowth functions. Our study reveals drastic differences in the behavior of proper finitely generated subalgebras and nonzero subideals. For instance, the growth of a proper finitely generated subalgebra H of a free Lie algebra L, with respect to any fixed free basis X, is exponentially small compared to the growth of the whole of L. Quite opposite, the cogrowth of any nonzero subideal S is exponentially small compared to the growth of L.

Cited by

• On the Hausdorff spectra of free pro-p groups and certain p-adic analytic groups

Related

• Growth of finitely generated simple Lie algebras
• Schreier's type formulae and two scales for growth of Lie algebras and groups
• Embeddings in Lie algebras of subexponential growth
• Subnormal subgroups in free groups, their growth and cogrowth
• How do algebras grow?