Representation of a Lie group

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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).

1 Representations on a complex finite-dimensional vector space
2 Representations on a finite-dimensional vector space over an arbitrary field
3 Representations on Hilbert spaces
4 Classification
5 Formulaic examples
6 References
7 See also

[edit] 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.

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.

[edit] 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.

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 an 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.

[edit] 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 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.

[edit] 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 give 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.

[edit] 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 $J = \left [ \begin{smallmatrix}0 & I_n \\ -I_n & 0\end{smallmatrix} \right ]$ , where I_n is the n×n identity matrix. Let

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

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 (n₁, …, 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 $\mathbb{F}_q$ .

[edit] References

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

Knapp, A. W. (2002). Lie groups beyond an introduction (2nd ed.) Boston:Birkhäuser.

Wulf Rossmann, Lie Groups: An Introduction Through Linear Groups (Oxford Graduate Texts in Mathematics), Oxford University Press ISBN 0-19-859683-9. The 2003 reprinting corrects some unfortunate typos.