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Categories: Lie groups | Representation theory

Representations of Lie groups

(Redirected from Representations of Lie groups/algebras)

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

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

Then $Sp_2(\mathbb{F}_q)$ is a symplectic group of rank n and is a finite group of Lie type. For $G = GL_n(\mathbb{F}_q)$ or $SL_n(\mathbb{F}_q)$ (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 $GL_n(\mathbb{F}_q)$ , then there exists a partition $(n_1,\ldots,n_r)\,\!$ 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

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

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 $\mathbb{F}_q$ .

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

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