Representations of Lie groups

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).

Formally, 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.

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.

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.

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

Formulaic examples

Let [itex]\mathbb{F}_q[itex] be a finite field of order q and characteristic p. Let [itex]G[itex] be a finite group of Lie type, that is, [itex]G[itex] is the [itex]\mathbb{F}_q[itex]-rational points of a connected reductive group [itex]\mathbb{G}[itex] defined over [itex]\mathbb{F}_q[itex]. For example, if n is a positive integer [itex]GL_n(\mathbb{F}_q)[itex] and [itex]SL_n(\mathbb{F}_q)[itex] are finite groups of Lie type. Let [itex]J = \begin{pmatrix}0 & I_n \\ -I_n & 0\end{pmatrix}[itex], where [itex]I_n\,\![itex] is the [itex]\,\!n \times n[itex] identity matrix. Let

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

Then [itex]Sp_2(\mathbb{F}_q)[itex] is a symplectic group of rank n and is a finite group of Lie type. For [itex]G = GL_n(\mathbb{F}_q)[itex] or [itex]SL_n(\mathbb{F}_q)[itex] (and some other examples), the standard Borel subgroup [itex]B\,\![itex] of [itex]G\,\![itex] is the subgroup of [itex]G\,\![itex] consisting of the upper triangular elements in [itex]G\,\![itex]. A standard parabolic subgroup of [itex]G\,\![itex] is a subgroup of [itex]G\,\![itex] which contains the standard Borel subgroup [itex]B\,\![itex]. If [itex]P\,\![itex] is a standard parabolic subgroup of [itex]GL_n(\mathbb{F}_q)[itex], then there exists a partition [itex](n_1,\ldots,n_r)\,\![itex] of [itex]n\,\![itex] (a set of positive integers [itex]n_j\,\![itex] such that [itex]n_1 + \ldots + n_r = n\,\![itex]) such that [itex]P = P_{(n_1,\ldots,n_r)} = M \times N[itex], where [itex]M \simeq GL_{n_1}(\mathbb{F}_q) \times \ldots \times GL_{n_r}(\mathbb{F}_q)[itex] has the form

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

and

[itex]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\}[itex],

where [itex]*\,\![itex] denotes arbitrary entries in [itex]\mathbb{F}_q[itex].

This section is still in progress. It should be finished soon.Vermi 01:32, 13 Apr 2005 (UTC)

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