On the group of polynomial antomorphisms of the affine planes. (Sur le groupe des automorphismes polynomiaux du plan affine.) (English) Zbl 1096.14046

Déserti, Julie

J. Algebra 297, No. 2, 584-599 (2006).

The main result of the paper under review is that, over a uncountable field \(k\) of characteristic 0, the automorphism group of the group \(\text{Aut}(k^2)\) of polynomial automorphisms of the affine plane is generated by the inner automorphisms of \(\text{Aut}(k^2)\) and the action of automorphisms of the field: If \(\varphi\) is an automorphism of the group \(\text{Aut}(k^2)\), then there exists a polynomial automorphism \(\psi\) and an automorphism \(\tau\) of the field \(k\) such that \(\varphi(f)=\tau(\psi f\psi^{-1})\) for all \(f\in \text{Aut}(k^2)\).
As an application, the author shows that an automorphism \(\varphi\) of \(\text{Aut}(k^2)\) is inner if and only if it preserves the determinant of the Jacobian matrix of the automorphisms of \(k^2\). Another application gives that the automorphisms of the semigroup \(\text{End}(k^2)\) of polynomial endomorphisms of \(k^2\) are compositions of inner automorphisms and automorphisms of the field.

Reviewer: Vesselin Drensky (Sofia)

MSC:

14R10	Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
13B25	Polynomials over commutative rings
14E07	Birational automorphisms, Cremona group and generalizations
58D05	Groups of diffeomorphisms and homeomorphisms as manifolds

Keywords:

polynomial automorphisms of affine plane; automorphisms of polynomial algebras; automorphisms of groups of automorphisms

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