View source for Algebraic group of transformations

An algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115201.png" /> acting regularly on an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115202.png" />. More precisely, it is a triplet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115203.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115204.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115205.png" />) is a morphism of algebraic varieties satisfying the conditions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115207.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115209.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152010.png" /> is the unit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152011.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152013.png" /> are defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152015.png" /> is called an algebraic group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152017.png" />-transformations. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152019.png" /> is the adjoint action or an action by shifts, is an algebraic group of transformations. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152020.png" /> is an algebraic subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152022.png" /> is its natural action on the affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152024.png" /> is an algebraic group of transformations. For each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152025.png" /> one denotes by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152026.png" /> the orbit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152027.png" />, and by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152028.png" /> the stabilizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152029.png" />. The orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152030.png" /> need not necessarily be closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152031.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152032.png" /> which is acting rationally (but not necessarily regularly) on an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152033.png" /> (this means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152034.png" /> is a rational mapping, and the above properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152035.png" /> are valid for ordinary points). It was shown by A. Weil [[#References|[3]]] that there always exists a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152036.png" />, birationally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152037.png" />, and such that the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152038.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152039.png" /> induced by the rational action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152040.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152041.png" /> is regular. The problem of describing the orbits, stabilizers, fields of invariant rational functions (cf. [[Invariants, theory of|Invariants, theory of]]), and of constructing quotient varieties are fundamental in the theory of algebraic groups of transformations and have numerous applications.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel,   "Linear algebraic groups" , Benjamin  (1969)  {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  J.A. Dieudonné,   "Invariant theory: old and new" , Acad. Press  (1971)  {{MR|0279102}} {{ZBL|0258.14011}} </TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  D. Mumford,   "Geometric invariant theory" , Springer  (1965)  {{MR|0214602}} {{ZBL|0147.39304}} </TD></TR><TR><TD valign="top">[2c]</TD> <TD valign="top">  D. Mumford,   "Projective invariants of projective structures and applications" , ''Proc. Internat. Congress mathematicians (Stockholm, 1962)'' , Inst. Mittag-Leffler  (1963)  pp. 526–530  {{MR|0175899}} {{ZBL|0154.20702}} </TD></TR><TR><TD valign="top">[2d]</TD> <TD valign="top">  C.S. Seshadri,   "Quotient spaces modulo reductive groupes and applications to moduli of vector bundles on algebraic curves" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars  (1971)  pp. 479–482  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Weil,   "On algebraic groups and homogeneous spaces"  ''Amer. J. Math.'' , '''77''' :  2  (1955)  pp. 355–391  {{MR|0074084}} {{ZBL|0065.14202}} </TD></TR></table>



====Comments====
The notion in question is also called an algebraic transformation space.