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In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras (indeed in the physics literature the distinction is often elided). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
Theoretical physics attempts to understand the world by making a model of reality, used for rationalizing, explaining, and predicting physical phenomena through a physical theory. There are three types of theories in physics: mainstream theories, proposed theories and fringe theories. ...
In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...
Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...
In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...
Representations on a complex finite-dimensional vector space
Let us first discuss representations acting on finite-dimensional complex vector spaces. A representation of a Lie group G on a finite-dimensional complex vector space V is a smooth group homomorphism Ψ:G→Aut(V) from G to the automorphism group of V. Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...
In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...
For n-dimensional V, the automorphism group of V is identified with a subset of complex square-matrices of order n. The automorphism group of V is given the structure of a smooth manifold using this identification. The condition that Ψ is smooth, in the definition above, means that Ψ is a smooth map from the smooth manifold G to the smooth manifold Aut(V).
If a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into GL(n,C). This is known as a matrix representation. In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ...
Representations on a finite-dimensional vector space over an arbitrary field A representation of a Lie group G on a vector space V (over a field K) is a smooth (i.e. respecting the differential structure) group homomorphism G→Aut(V) from G to the automorphism group of V. If a basis for the vector space V is chosen, the representation can be expressed as a homomorphism into GL(n,K). This is known as a matrix representation. Two representations of G on vector spaces V, W are equivalent if they have the same matrix representations with respect to some choices of bases for V and W. Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...
In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, a smooth function is one that is infinitely differentiable, i. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...
In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ...
On the Lie algebra level, there is a corresponding linear mapping from the Lie algebra of G to End(V) preserving the Lie bracket [ , ]. See representation of Lie algebras for the Lie algebra theory. In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ...
A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
In mathematics, if φ: G→H is a homomorphism of Lie groups, and g and h are the Lie algebras of G and H respectively, then the induced map φ* on tangent spaces is a homomorphism of Lie algebras, i. ...
If the homomorphism is in fact an monomorphism, the representation is said to be faithful. In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ...
A unitary representation is defined in the same way, except that G maps to unitary matrices; the Lie algebra will then map to skew-hermitian matrices. In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group...
In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse...
In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A* is also its negative. ...
If G is a compact Lie group, every finite-dimensional representation is equivalent to a unitary one.
Representations on Hilbert spaces A representation of a Lie group G on a complex Hilbert space V is a group homomorphism Ψ:G → B(V) from G to B(V), the group of bounded linear operators of V which have a bounded inverse, such that the map G x V → V given by (g,v) → Ψ(g) v is continuous. Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...
In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...
In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
This definition can handle representations on infinite-dimensional Hilbert spaces. Such representations can be found in e.g. quantum mechanics, but also in Fourier analysis as shown in the following example.
Let G=R, and let the complex Hilbert space V be L^2(R). We define the representation Ψ:R → B(L^2(R)) by (Ψ(r)f)(x) → f(r^{-1} x).
See also Wigner's classification for representations of the Poincaré group. In mathematics and theoretical physics, Wigners classification is a classification of the nonnegative energy irreducible unitary representations of the Poincaré group, which have sharp mass eigenvalues. ...
In physics and mathematics, the Poincaré group is the group of isometries of Minkowski spacetime. ...
Classification If G is a semisimple group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations. The irreducibles are indexed by highest weight; the allowable (dominant) highest weights satisfy a suitable positivity condition. In particular, there exists a set of fundamental weights, indexed by the vertices of the Dynkin diagram of G, such that dominant weights are simply non-negative integer linear combinations of the fundamental weights. In mathematics, the term semisimple is used in a number of related ways, within different subjects. ...
In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...
In mathematics, the term irreducible is used in several ways. ...
Given a set S of complex matrices, each of which is diagonalizable and any two of which commute under multiplication, it is always possible to diagonalize all the elements of S simultaneously. ...
See also Simple Lie group. ...
If G is a commutative compact Lie group, then its irreducible representations are simply the continuous characters of G: see Pontryagin duality for this case. In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...
In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ...
A quotient representation is a quotient module of the group ring. If B is a submodule of a module A of a ring R, then the quotient space A/B is also a module of R. Its called the quotient module. ...
In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described...
Formulaic examples Let be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the -rational points of a connected reductive group defined over . For example, if n is a positive integer and are finite groups of Lie type. Let , where is the identity matrix. Let . Then is a symplectic group of rank n and is a finite group of Lie type. For or (and some other examples), the standard Borel subgroup of is the subgroup of consisting of the upper triangular elements in . A standard parabolic subgroup of is a subgroup of which contains the standard Borel subgroup . If is a standard parabolic subgroup of , then there exists a partition of (a set of positive integers such that ) such that , where has the form , and , where denotes arbitrary entries in .
References Knapp, A. W. (2002). Lie groups beyond an introduction (2nd ed.) Boston:Birkhäuser.
Rossmann, W. (2002). Lie groups : an introduction through linear groups. Oxford:Oxford Univ. Press.
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