Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points

ABSTRACT

Let F _m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F _m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F _m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) ^G is not finitely generated.

Key Words:

• Automorphisms of free algebras
• Left nilpotent Leibniz algebras
• Tame automorphisms
• Wild automorphisms

Mathematics Subject Classification:

• Primary 17A32
• Secondary 17A36
• 17A50

ACKNOWLEDGMENT

The work is partially supported by Grant MM1106/2001 of the Bulgarian Foundation for Scientific Research. Part of this project was carried out when he visited the Aristotle University of Thessaloniki. He is very grateful for the creative atmosphere and the warm hospitality. The authors are thankful to the referee for the second part of Remarks 4.6.

Notes

^#Communicated by I. Shestakov.