Involution

An endomorphism of order two, that is, a mapping of an object onto itself whose square is the identity morphism (see also Category with involution). A periodic mapping, that is, a morphism some non-zero power of which is the identity morphism, is also sometimes called an involution. The minimum of such powers is called the period of the involution.

Often, by the involutions of a group $ G $ one means the elements of order two in $ G $.

An involution in an algebra $ E $ over the field of real or complex numbers is a mapping $ x \rightarrow x ^ {*} $ of $ E $ onto itself satisfying the following involution axioms: 1) $ x ^ {**} = x $ for all $ x \in E $; 2) $ ( x + y ) ^ {*} = x ^ {*} + y ^ {*} $ for all $ x , y \in E $; 3) $ ( \lambda x ) ^ {*} = \overline \lambda \; x ^ {*} $ for all $ x \in E $ and for all $ \lambda $ in the corresponding field; and 4) $ ( x y ) ^ {*} = y ^ {*} x ^ {*} $ for all $ x , y \in E $. An algebra $ E $ over the complex field endowed with an involution is called a symmetric algebra or involution algebra.

References

[1]	P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964)

An involution in projective geometry is a projective transformation whose square is the identity transformation. A non-identity involution of the real projective line has just two fixed points (a hyperbolic involution) or has no fixed points (an elliptic involution). If $ A $ and $ B $ are the fixed points of a hyperbolic involution, then the points $ M $ and $ M _ {1} $ corresponding to them harmonically divide the pair $ A , B $. Every involution on the projective plane is a hyperbolic homology.

References

[1]	N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)

An involution of an algebraic variety is an automorphism of the variety of order two. If $ X $ is a non-singular projective algebraic variety over an algebraically closed field $ k $ and $ g $ is an involution of $ X $, then the quotient variety $ X / \{ g \} $ with respect to the action of the cyclic group $ \{ g \} $ is a projective variety, called the quotient under the involution $ g $. The set of fixed points $ F ( g) $ of $ g $ forms a non-singular subvariety of $ X $. If $ F ( g) $ has codimension 1 at each point, the image of $ g $ is a non-singular variety. The numerical invariants of a non-singular model $ \overline{X}\; $ of the variety $ X / \{ g \} $ can be calculated by means of the Lefschetz formula.

References

[1]	M.F. Atiyah, I.M. Singer, "The index of elliptic operators III" Ann. of Math. (2) , 87 : 1 (1968) pp. 546–604
[2]	I.V. Dolgachev, V.A. Iskovskikh, "Geometry of algebraic varieties" J. Soviet Math. , 5 : 6 (1976) pp. 803–864 Itogi Nauk. Algebra Topol. Geom. , 12 (1975) pp. 77–170
[3]	L. Godeaux, "Les involutions cycliques appartenant à une surface algébrique" , Hermann (1963)