Abstract:
We define a class of ◦-varieties of algebras and prove that the
tame automorphism group of a free algebra of rank two of any ◦-variety
of algebras over a field admits an amalgamated free product structure.
In particular, the automorphism group of a free right-symmetric algebra
of rank two admits an amalgamated free product structure. Using this
structure, we prove that any locally finite group of automorphisms of this
algebra is conjugate to a subgroup of affine or triangular automorphisms.
This implies that any reductive group of automorphisms of a two-generated free right-symmetric algebra is linearizable and any locally nilpotent
derivation of this algebra is triangulable over a field of characteristic zero.
All of these results are true for free commutative and free non-associative
algebras of rank two.
