In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.
In the following we will assume all groups are Hausdorff spaces.
Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include
the circle group T and the torus groups Tn,
the orthogonal group O(n), the special orthogonal group SO(n) and its covering spin group Spin(n),
the unitary group U(n) and the special unitary group SU(n),
the compact forms of the exceptional Lie groups: G2, F4, E6, E7, and E8.
The classification theorem of compact Lie groups states that up to finite extensions and finite covers this exhausts the list of examples (which already includes some redundancies). This classification is described in more detail in the next subsection.
Given any compact Lie group G one can take its identity component G0, which is connected. The quotient group G/G0 is the group of components π0(G) which must be finite since G is compact. We therefore have a finite extension
Meanwhile, for connected compact Lie groups, we have the following result:
Theorem: Every connected compact Lie group is the quotient by a finite central subgroup of a product of a simply connected compact Lie group and a torus.
Thus, the classification of connected compact Lie groups can in principle be reduced to knowledge of the simply connected compact Lie groups together with information about their centers. (For information about the center, see the section below on fundamental group and center.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex
Lie groups are manifolds with a group structure. The interaction between the geometric and the algebraic structure of these objects gives rise to a rich and beautiful subject with various applications
The goal of the course is to introduce relativistic quantum field theory as the conceptual and mathematical framework describing fundamental interactions.
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group G is a compact, connected, abelian Lie subgroup of G (and therefore isomorphic to the standard torus Tn). A maximal torus is one which is maximal among such subgroups. That is, T is a maximal torus if for any torus T′ containing T we have T = T′. Every torus is contained in a maximal torus simply by dimensional considerations.
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).
DISPLAYTITLE:G2 (mathematics) In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G2 has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14. The compact form of G2 can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional real spinor representation (a spin representation).
This paper presents a self contained approach to the theory of convolution operators on locally compact groups (both commutative and non commutative) based on the use of the Figà–Talamanca Herz algebr
We present a self-contained proof of the following famous extension theorem due to Carl Herz. A closed subgroup H of a locally compact group G is a set of p-synthesis in G if and only if, for every u
Quantum Field Theories (QFTs) in Anti-de Sitter (AdS) spacetime are often strongly coupled when the radius of AdS is large, and few methods are available to study them. In this work, we develop a Hami