Projective group

Projective group
Jgmoxness · CC BY-SA 3.0 · source
Name	Projective group
Type	Group
Field	Mathematics
Related	Projective linear group; Möbius group; PGL; PSL

Projective group The projective group is a family of groups acting on projective spaces and arising in algebraic geometry, group theory, and differential geometry, with connections to classical works by Felix Klein, Évariste Galois, Henri Poincaré, Bernhard Riemann and David Hilbert. It appears in the study of symmetries of Riemann surfaces, Complex projective space, Real projective space, Algebraic varietys and in the classification of Lie groups, linking topics treated in texts by Élie Cartan, Hermann Weyl, Emmy Noether and John von Neumann.

Definition and basic properties

In algebraic terms the projective group is frequently presented as a quotient of a general linear group such as GL(n+1) by its center, producing groups related to PGL(n+1) or PSL(n+1), with foundational precedents in the work of Augustin-Louis Cauchy, Joseph-Louis Lagrange, Carl Friedrich Gauss and Arthur Cayley. Fundamental properties include transitivity on flags in Pn and relations to stabilizer subgroups studied by Sophus Lie, Wilhelm Killing and Élie Cartan. Other basic facts about these groups are analogous to results in the theory of Simple groups, semisimple structures, and central isogenies considered by Claude Chevalley and Robert Langlands.

Examples and classical groups

Concrete instances include the real projective group associated to RP^n linked to O(n+1), the complex projective group tied to CP^n and U(n+1), and finite projective groups exemplified by PSL(2,q) where q is a power of a prime, appearing in the classification of Finite simple groups and in papers by Émile Mathieu and Évariste Galois. Other classical examples are connected to Sp(2n), E_6 and E_7 through their actions on projective varieties studied by Alexander Grothendieck and David Mumford.

Actions on projective spaces

Projective groups act by collineations on Pn preserving incidence relations studied since Jean-Victor Poncelet, Plücker and Julius Plücker and later formalized in the Erlangen Program of Felix Klein. These actions are central in rigidity results connected to Mostow and to dynamics investigated by George D. Birkhoff, Andrey Kolmogorov and Marston Morse. In algebraic geometry contexts their actions on Grassmannians, on moduli spaces like Mg and on flag varieties are used in work of Alexander Grothendieck, Pierre Deligne and Nicholas Katz.

Algebraic and Lie theoretic structure

As algebraic groups the projective group inherits structure from linear algebraic groups and their Lie algebras, with classification guided by Cartan and Dynkin diagram techniques used by Élie Cartan and Nikolai Chebotaryov. Over local fields the groups relate to p-adic groups studied by John Tate and Jean-Pierre Serre; over global fields they enter the scope of the Langlands program formulated by Robert Langlands and elaborated by Pierre Deligne and Michael Harris. Central extensions, covering groups, and double covers refer back to constructions of Issai Schur and Stefan Banach.

Homographies and projective transformations

Projective transformations, or homographies, are represented by nonsingular matrices modulo scalars, a viewpoint developed in the classical literature of August Ferdinand Möbius, Carl Gustav Jacobi, Joseph Liouville and Sophus Lie. These maps include special cases such as Möbius transformations on the Riemann sphere studied by Bernhard Riemann and Paul Koebe, and real collineations relevant to the Poncelet porism and to geometric problems treated by J. J. Sylvester and Arthur Cayley. Algebraic descriptions connect to invariant theory pursued by David Hilbert and Emmy Noether.

Applications in geometry and topology

Projective groups appear in the study of geometric structures on manifolds investigated by William Thurston, Dennis Sullivan, Geoffrey Mess and Yair Minsky and feature in geometric topology questions linked to Kleinian group theory, hyperbolic geometry from Henri Poincaré and moduli problems treated by Max Delbrück. They also underlie constructions in algebraic topology appearing in the work of Raoul Bott, Michael Atiyah and Isadore Singer through index theorems and characteristic classes, and in differential geometry in contributions by Shing-Tung Yau and Richard S. Hamilton.

Representations and cohomology of projective groups

Representation theory of projective groups intersects the work of Hermann Weyl, Harish-Chandra, George Lusztig and David Vogan and plays a role in harmonic analysis on homogeneous spaces studied by Elias M. Stein and Fritz John. Cohomological methods applied to these groups draw on the theories of Jean-Pierre Serre and Alexander Grothendieck and connect to automorphic forms in the program of Robert Langlands, with implications for arithmetic following investigations by Gerd Faltings and Andrew Wiles.

Category:Algebraic groups