Alle productspecificaties Samenvatting Algebraic groups and Lie groups are important in most major areas of mathematics, occuring in diverse roles such as the symmetries of differential equations and as central figures in the Langlands program for number theory.
Overview[ edit ] The set of all complex numbers with absolute value 1 corresponding to points on the circle of center 0 and radius 1 in the complex plane is a Lie group under complex multiplication: Lie groups are smooth differentiable manifolds and as such can be studied using differential calculusin contrast with the case of more general topological groups.
One of the key ideas in the theory of Lie groups is to replace the global object, the group, with its local or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra.
Lie groups play an enormous role in modern geometryon several different levels.
Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E 3 of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal groupwhereas in projective geometry one is interested in the properties invariant under the projective group.
This idea later led to the notion of a G-structurewhere G is a Lie group of "local" symmetries of a manifold. Lie groups and their associated Lie algebras play a major role in modern physics, with the Lie group typically playing the role of a symmetry of a physical system.
Here, the representations of the Lie group or of its Lie algebra are especially important. Representation theory is used extensively in particle physics. On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifoldthis action provides a measure of rigidity and yields a rich algebraic structure.
The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory.
In the s—s, Ellis KolchinArmand Boreland Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field.
This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groupsas well as in algebraic geometry.
The theory of automorphic formsan important branch of modern number theorydeals extensively with analogues of Lie groups over adele rings ; p-adic Lie groups play an important role, via their connections with Galois representations in number theory.
Definitions and examples[ edit ] A real Lie group is a group that is also a finite-dimensional real smooth manifoldin which the group operations of multiplication and inversion are smooth maps.In mathematics, a Lie group (pronounced / l iː / "Lee") is a group that is also a differentiable manifold, with the property that the group operations are rutadeltambor.com groups are named after Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups..
In rough terms, a Lie group is a continuous group, that is, one whose elements are described. Aug 15, · Essays in the History of Lie Groups and Algebraic Groups by Armand Borel, , available at Book Depository with free delivery worldwide.4/4(3). [This is the third part of a four part essay–here is Part I.].
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Essays in the history of Lie groups and algebraic groups. [Armand Borel] -- Lie groups and algebraic groups are important in many major areas of mathematics and mathematical physics. We find them in diverse roles, notably as groups of automorphisms of geometric structures. Writing Mini-Lessons: Narrative Engaging Beginnings/Leads.
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