On tame and wild automorphisms of algebras. (English. Russian original) Zbl 1339.16026
J. Math. Sci., New York 206, No. 6, 660-667 (2015); translation from Fundam. Prikl. Mat. 18, No. 4, 79-88 (2013).
Let \(A_n\) be the \(n\)-generated free associative algebra (the polynomial algebra in \(n\) non-commuting variables) over a field \(F\) and let \(R\) be the ideal of \(A_n\) consisting of all polynomials without constant term. It is clear that, up to an affine automorphism, all automorphisms of \(A_n\) can be moved to automorphisms of \(R\) which act identically on \(R/R^2\), preserving the property of being tame or wild.
In the paper under review the authors study such automorphisms of \(R\). They give a canonical form modulo \(R^3\) and \(R^4\), up to a tame automorphism, of the possible wild automorphisms of \(A_n\). Then they consider the Anick automorphism \(\delta=(x+z(xz-zy),y+(xz-zy)z,z)\) of \(A_3=F\langle x,y,z\rangle\) which is known to be wild [U. U. Umirbaev, J. Reine Angew. Math. 605, 165-178 (2007; Zbl 1126.16021)]. The opinion of the reviewer is that the main result of the paper is that there are tame automorphisms which act in the same way as the Anick automorphism on \(R\) modulo \(R^{17}\).
The paper raises the problem for the approximation of wild automorphisms of \(A_n\) by tame automorphisms. By [D. J. Anick, J. Algebra 82, 459-468 (1983; Zbl 0535.13014)], for every endomorphism with invertible Jacobian matrix \(\varphi\) of the polynomial algebra \(F[x_1,\ldots,x_n]\) there exists a sequence of tame automorphisms \(\varphi_k\), \(k=1,2,\ldots\), such that \(\varphi\) and \(\varphi_k\) act in the same way modulo the \(k\)-th power of the augmentation ideal of \(F[x_1,\ldots,x_n]\) consisting of polynomials without constant term. This means that, with respect to the formal power series topology, the group of tame automorphisms is dense in the set of endomorphisms with invertible Jacobian matrix. Translated in the language of Anick, the problem raised in the present paper can be stated as: Is the subgroup of tame automorphisms of \(A_n\) dense in the whole automorphism group with respect to the formal power series topology on \(A_n\)?
In the paper under review the authors study such automorphisms of \(R\). They give a canonical form modulo \(R^3\) and \(R^4\), up to a tame automorphism, of the possible wild automorphisms of \(A_n\). Then they consider the Anick automorphism \(\delta=(x+z(xz-zy),y+(xz-zy)z,z)\) of \(A_3=F\langle x,y,z\rangle\) which is known to be wild [U. U. Umirbaev, J. Reine Angew. Math. 605, 165-178 (2007; Zbl 1126.16021)]. The opinion of the reviewer is that the main result of the paper is that there are tame automorphisms which act in the same way as the Anick automorphism on \(R\) modulo \(R^{17}\).
The paper raises the problem for the approximation of wild automorphisms of \(A_n\) by tame automorphisms. By [D. J. Anick, J. Algebra 82, 459-468 (1983; Zbl 0535.13014)], for every endomorphism with invertible Jacobian matrix \(\varphi\) of the polynomial algebra \(F[x_1,\ldots,x_n]\) there exists a sequence of tame automorphisms \(\varphi_k\), \(k=1,2,\ldots\), such that \(\varphi\) and \(\varphi_k\) act in the same way modulo the \(k\)-th power of the augmentation ideal of \(F[x_1,\ldots,x_n]\) consisting of polynomials without constant term. This means that, with respect to the formal power series topology, the group of tame automorphisms is dense in the set of endomorphisms with invertible Jacobian matrix. Translated in the language of Anick, the problem raised in the present paper can be stated as: Is the subgroup of tame automorphisms of \(A_n\) dense in the whole automorphism group with respect to the formal power series topology on \(A_n\)?
Reviewer: Vesselin Drensky (Sofia)
MSC:
| 16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |
| 16W20 | Automorphisms and endomorphisms |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
| 14E07 | Birational automorphisms, Cremona group and generalizations |
| 14R15 | Jacobian problem |
Keywords:
free associative algebras; tame automorphisms; wild automorphisms; Anick automorphism; automorphism groups; monic automorphisms; Jacobian matrixReferences:
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