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3.3 (1969)

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(Well-known problem). Describe the automorphism group of the free associative algebra with $n$ generators, $n \geqslant 2$.

Contributor: L. A. Bokut’

Progress

It is known that all automorphisms are tame for $n = 2$ (A. J. Czerniakiewicz, Trans. Amer. Math. Soc., 160 (1971), 393–401; 171 (1972), 309–315; L. G. Makar-Limanov, Funct. Anal. Appl., 4 (1971), 262–264). For $n = 3$, the Anick automorphism is wild (U. U. Umirbaev, J. Reine Angew. Math., 605 (2007), 165–178). There is an algorithm for recognizing automorphisms among endomorphisms of free associative algebras of finite rank over constructive fields (A. V. Jagzhev, Siberian. Math. J., 21 (1980), 142–146).

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