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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. 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×VV 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 L2(R). We define the representation Ψ:R → B(L2(R)) by Ψ(r){f(x)} → f(r−1x).

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,Fq) is a symplectic group of rank n and is a finite group of Lie type. For G = GL(n, Fq) or SL(n, Fq) (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, Fq), then there exists a partition (n1, …, nr) 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.

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