On Generators of the Tame Automorphism Group of Free Metabelian Lie Algebras

Abstract

We prove that the tame automorphism group TAut(M _n ) of a free metabelian Lie algebra M _n in n variables over a field k is generated by a single nonlinear automorphism modulo all linear automorphisms if n ≥ 4 except the case when n = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M ₄) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M ₃) cannot be generated by any finite number of automorphisms modulo all linear automorphisms.

Key Words:

• Automorphisms
• Elementary reductions
• Free Lie algebras
• Metabelian algebras

2010 Mathematics Subject Classification:

• Primary: 17B01
• 17B40
• Secondary: 17B30
• 17A36

ACKNOWLEDGMENTS

I am grateful to Department of Mathematics of Wayne State University for hospitality and excellent working conditions, where part of this work has been done. I am also grateful to Professor U. Umirbaev for helpful discussions and comments.

Notes

Communicated by I. Shestakov.