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# Representation of a Lie group

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. The physics literature sometimes passes over the distinction between Lie groups and Lie algebras.[1]

Representations on a complex finite-dimensional vector spaceIf a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into general linear group GL(n,C). This is known as a matrix representation.

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*.

For *n*-dimensional *V*, the automorphism group of *V* is identified with a subset of the 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 general linear group 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 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 general linear group 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 [ , ]. See representation 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 to skew-hermitian matrices.

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*×*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 the Poincaré group.

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. The characters of the irreducible representations are given by the Weyl character formula.

If G is a commutative Lie group, then its irreducible representations are simply the continuous characters of G: see Pontryagin duality for this case.

A quotient representation is a quotient module of the group ring.

Formulaic examples

Let Fq be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the Fq-rational points of a connected reductive group G defined over Fq. For example, if n is a positive integer GL(n, Fq) and SL(n, Fq) are finite groups of Lie type. Let \( J = \left [ \begin{smallmatrix}0 & I_n \\ -I_n & 0\end{smallmatrix} \right ] \), where In is the n×n identity matrix. Let

\(Sp_2(\mathbb{F}_q) = \left \{ g \in GL_{2n}(\mathbb{F}_q) | ^tgJg = J \right \}.\)

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 \(P = P_{(n_1,\ldots,n_r)} = M \times N\), where \(M \simeq GL_{n_1}(\mathbb{F}_q) \times \ldots \times GL_{n_r}(\mathbb{F}_q)\) has the form

\( M = \left \{\left.\begin{pmatrix}A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & A_r\end{pmatrix}\right|A_j \in GL_{n_j}(\mathbb{F}_q), 1 \le j \le r \right \},\)

and

\( N=\left \{\begin{pmatrix}I_{n_1} & * & \cdots & * \\ 0 & I_{n_2} & \cdots & * \\ \vdots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & I_{n_r}\end{pmatrix}\right\},\)

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

See also

Representation theory of the Lorentz group

Representation theory of Hopf algebras

Adjoint representation of a Lie group

List of Lie group topics

Symmetry in quantum mechanics

References

Hall 2003 Chapter 2.

Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6

Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, ISBN 0-387-40122-9.

Knapp, Anthony W. (2002), Lie Groups Beyond an Introduction, Progress in Mathematics 140 (2nd ed.), Boston: Birkhäuser.

Rossmann, Wulf (2001), Lie Groups: An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford University Press, ISBN 978-0-19-859683-7. The 2003 reprint corrects several typographical mistakes.

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

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