Algebraic group of transformations

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An algebraic group $G$ acting regularly on an algebraic variety $V$ . More precisely, it is a triplet $(G,\ V,\ \tau )$ where $\tau : \ G \times V \rightarrow V$ ( $\tau (g,\ x ) = gx$ ) is a morphism of algebraic varieties satisfying the conditions: $ex = x$ , $g(hx) = (gh)x$ for all $x \in V$ and $g,\ h \in G$ (where $e$ is the unit of $G$ ). If $G,\ V$ and $\tau$ are defined over a field $k$ , then $( G,\ V,\ \tau )$ is called an algebraic group of $k$ -transformations. For instance, $( G,\ G,\ \tau )$ , where $\tau$ is the adjoint action or an action by shifts, is an algebraic group of transformations. If $G$ is an algebraic subgroup in $\mathop{\rm GL}\nolimits (n)$ and $\tau$ is its natural action on the affine space $V = k ^{n}$ , then $(G,\ V,\ \tau )$ is an algebraic group of transformations. For each point $x \in V$ one denotes by $G(x) = \{ {gx} : {g \in G} \}$ the orbit of $x$ , and by $G _{x} = \{ {g \in G} : {gx = x} \}$ the stabilizer of $x$ . The orbit $G(x)$ need not necessarily be closed in $V$ , but closed orbits exist always, e.g. orbits of minimal dimension are closed. An algebraic group of transformations is sometimes understood to mean a group $G$ which is acting rationally (but not necessarily regularly) on an algebraic variety $V$ (this means that $\tau : \ G \times V \rightarrow V$ is a rational mapping, and the above properties of $\tau$ are valid for ordinary points). It was shown by A. Weil [3] that there always exists a variety $V ^{1}$ , birationally isomorphic to $V$ , and such that the action of $G$ on $V ^{1}$ induced by the rational action of $G$ on $V$ is regular. The problem of describing the orbits, stabilizers, fields of invariant rational functions (cf. Invariants, theory of), and of constructing quotient varieties are fundamental in the theory of algebraic groups of transformations and have numerous applications.

References

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