### A classification of reductive linear groups.

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The main purpose of this article is to give an explicit algebraic action of the group ${S}_{3}$ of permutations of 3 elements on affine four-dimensional complex space which is not conjugate to a linear action.

2000 Mathematics Subject Classification: Primary: 14R10. Secondary: 14R20, 13N15.Let R be a UFD containing a field of characteristic 0, and Bm = R[Y1, . . . , Ym] be a polynomial ring over R. It was conjectured in [5] that if D is an R-elementary monomial derivation of B3 such that ker D is a finitely generated R-algebra then the generators of ker D can be chosen to be linear in the Yi ’s. In this paper, we prove that this does not hold for B4. We also investigate R-elementary derivations D of Bm...

A concept of a slice of a semisimple derivation is introduced. Moreover, it is shown that a semisimple derivation d of a finitely generated commutative algebra A over an algebraically closed field of characteristic 0 is nothing other than an algebraic action of a torus on Max(A), and, using this, that in some cases the derivation d is linearizable or admits a maximal invariant ideal.

A. Crachiola and L. Makar-Limanov [J. Algebra 284 (2005)] showed the following: if X is an affine curve which is not isomorphic to the affine line ${\xb9}_{k}$, then ML(X×Y) = k[X]⊗ ML(Y) for every affine variety Y, where k is an algebraically closed field. In this note we give a simple geometric proof of a more general fact that this property holds for every affine variety X whose set of regular points is not k-uniruled.

We give a short proof of a counterexample (due to Daigle and Freudenburg) to Hilbert's fourteenth problem in dimension five.

We survey counterexamples to Hilbert’s Fourteenth Problem, beginning with those of Nagata in the late 1950s, and including recent counterexamples in low dimension constructed with locally nilpotent derivations. Historical framework and pertinent references are provided. We also include 8 important open questions.

It is explained that the following two problems are equivalent: (i) describing all affine rulings of any given weighted projective plane; (ii) describing all weighted-homogeneous locally nilpotent derivations of k[X,Y,Z]. Then the solution of (i) is sketched. (Outline of our joint work with Peter Russell.)

We classify all finitely generated integral algebras with a rational action of a reductive group such that any invariant subalgebra is finitely generated. Some results on affine embeddings of homogeneous spaces are also given.

We consider the family of polynomials in $\mathbf{C}[x,y,z]$ of the form ${x}^{2}y-{z}^{2}-xq(x,z)$. Two such polynomials ${P}_{1}$ and ${P}_{2}$ are equivalent if there is an automorphism ${\varphi}^{*}$ of $\mathbf{C}[x,y,z]$ such that ${\varphi}^{*}\left({P}_{1}\right)={P}_{2}$. We give a complete classification of the equivalence classes of these polynomials in the algebraic and analytic category. As a consequence, we find the following results. There are explicit examples of inequivalent polynomials ${P}_{1}$ and ${P}_{2}$ such that the zero set of ${P}_{1}+c$ is isomorphic to the zero set of ${P}_{2}+c$ for all $c\in \mathbf{C}$. There exist polynomials which are algebraically...

Let $G\to GL\left(V\right)$ be a representation of a reductive linear algebraic group $G$ on a finite-dimensional vector space $V$, defined over an algebraically closed field of characteristic zero. The categorical quotient $X=V//G$ carries a natural stratification, due to D. Luna. This paper addresses the following questions:(i) Is the Luna stratification of $X$ intrinsic? That is, does every automorphism of $V//G$ map each stratum to another stratum?(ii) Are the individual Luna strata in $X$ intrinsic? That is, does every automorphism...

We give a new proof of Miyanishi's theorem on the classification of the additive group scheme actions on the affine plane.

We show that every automorphism of the group ${\mathcal{G}}_{n}:=Aut\left({\mathbb{A}}^{n}\right)$ of polynomial automorphisms of complex affine $n$-space ${\mathbb{A}}^{n}={\u2102}^{n}$ is inner up to field automorphisms when restricted to the subgroup $T{\mathcal{G}}_{n}$ of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension $n=2$ where all automorphisms are tame: $T{\mathcal{G}}_{2}={\mathcal{G}}_{2}$. The methods are different, based on arguments from algebraic group actions.

Let G be a complex affine algebraic group and H,F ⊂ G be closed subgroups. The homogeneous space G/H can be equipped with the structure of a smooth quasiprojective variety. The situation is different for double coset varieties F∖∖G//H. We give examples showing that the variety F∖∖G//H does not necessarily exist. We also address the question of existence of F∖∖G//H in the category of constructible spaces and show that under sufficiently general assumptions F∖∖G//H does exist as a constructible space....

Let X be an affine toric variety. The total coordinates on X provide a canonical presentation $$\overline{X}\to X$$ of X as a quotient of a vector space $$\overline{X}$$ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.

L. Makar-Limanov, P. van Rossum, V. Shpilrain and J.-T. Yu solved the stable equivalence problem for the polynomial ring k[x,y] when k is a field of characteristic 0. In this note we give an affirmative solution for an arbitrary field k.

2010 Mathematics Subject Classification: 14L99, 14R10, 20B27.If F is a polynomial automorphism over a finite field Fq in dimension n, then it induces a permutation pqr(F) of (Fqr)n for every r О N*. We say that F can be “mimicked” by elements of a certain group of automorphisms G if there are gr О G such that pqr(gr) = pqr(F). Derksen’s theorem in characteristic zero states that the tame automorphisms in dimension n і 3 are generated by the affine maps and the one map (x1+x22, x2,ј, xn). We show...