- Representation of a Lie group
In

mathematics andtheoretical physics , the idea of a**representation of a**plays an important role in the study of continuousLie group symmetry . A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations ofLie algebra s (indeed in the physics literature the distinction is often elided).**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 complexvector space "V" is a smoothgroup homomorphism Ψ:"G"→Aut("V") from "G" to theautomorphism group of "V".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".**Representations on a finite-dimensional vector space over an arbitrary field**A representation of a

Lie group "G" on avector space "V" (over a field "K") is a smooth (i.e. respecting the differential structure)group homomorphism "G"→Aut("V") from "G" to theautomorphism 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".On the Lie algebra level, there is a corresponding linear mapping from the Lie algebra of G to End("V") preserving the

Lie bracket [ , ] . Seerepresentation of Lie algebras for the Lie algebra theory.If the homomorphism is in fact a

monomorphism , the representation is said to be "faithful".A

unitary representation is defined in the same way, except that G maps to unitary matrices; the Lie algebra will then map toskew-hermitian matrices.If "G" is a compact Lie group, every finite-dimensional representation is equivalent toa unitary one.

**Representations on Hilbert spaces**A representation of a

Lie group "G" on a complexHilbert space "V" is agroup 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"×"V" → "V" given by ("g","v") → Ψ("g")"v" is continuous.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 thePoincaré group .**Classification**If G is a

semisimple group, its finite-dimensional representations can be decomposed asdirect sum s ofirreducible representation s. 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 theDynkin diagram of G, such that dominant weights are simply non-negative integer linear combinations of the fundamental weights. The characters of the irreducible representations are given by theWeyl character formula .If G is a commutative

Lie group , then its irreducible representations are simply the continuous characters of G: seePontryagin duality for this case.A quotient representation is a

quotient module of thegroup ring .**Formulaic examples**Let

**F**_{"q"}be a finite field of order "q" and characteristic "p". Let "G" be a finite group of Lie type, that is, "G" is the**F**_{"q"}-rational points of a connected reductive group "G" defined over**F**_{"q"}. For example, if "n" is a positive integer GL("n",**F**_{"q"}) and SL(n,**F**_{"q"}) are finite groups of Lie type. Let $J\; =\; left\; [\; egin\{smallmatrix\}0\; I\_n\; \backslash \; -I\_n\; 0end\{smallmatrix\}\; ight\; ]$, where "I"_{n}is the "n"×"n" identity matrix. Let: $Sp\_2(mathbb\{F\}\_q)\; =\; left\; \{\; g\; in\; GL\_\{2n\}(mathbb\{F\}\_q)\; |\; ^tgJg\; =\; J\; ight\; \}.$

Then Sp(2,

**F**_{"q"}) is a symplectic group of rank "n" and is a finite group of Lie type. For "G" = GL("n",**F**_{"q"}) or SL("n",**F**_{"q"}) (and some other examples), the "standard Borel subgroup " "B" of "G" is the subgroup of "G" consisting of the upper triangular elements in "G". A "standard parabolic subgroup " of "G" is a subgroup of "G" which contains the standard Borel subgroup "B". If "P" is a standard parabolic subgroup of GL("n",**F**_{"q"}), then there exists a partition ("n"_{1}, …, "n"_{r}) of "n" (a set of positive integers $n\_j,!$ such that $n\_1\; +\; ldots\; +\; n\_r\; =\; n,!$) such that $P\; =\; P\_\{(n\_1,ldots,n\_r)\}\; =\; M\; imes\; N$, where $M\; simeq\; GL\_\{n\_1\}(mathbb\{F\}\_q)\; imes\; ldots\; imes\; GL\_\{n\_r\}(mathbb\{F\}\_q)$ has the form: $M\; =\; left\; \{left.egin\{pmatrix\}A\_1\; 0\; cdots\; 0\; \backslash \; 0\; A\_2\; cdots\; 0\; \backslash \; vdots\; ddots\; ddots\; vdots\; \backslash \; 0\; cdots\; 0\; A\_rend\{pmatrix\}\; ight|A\_j\; in\; GL\_\{n\_j\}(mathbb\{F\}\_q),\; 1\; le\; j\; le\; r\; ight\; \},$

and

: $N=left\; \{egin\{pmatrix\}I\_\{n\_1\}\; *\; cdots\; *\; \backslash \; 0\; I\_\{n\_2\}\; cdots\; *\; \backslash \; vdots\; ddots\; ddots\; vdots\; \backslash \; 0\; cdots\; 0\; I\_\{n\_r\}end\{pmatrix\}\; ight\},$

where $*,!$ denotes arbitrary entries in $mathbb\{F\}\_q$.

**ee also***

Lie algebra representation

*Representation theory of Hopf algebras

*Adjoint representation

*List of Lie group topics **References***

*.

*.

*. The 2003 reprint corrects several typographical mistakes.

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