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Representations of Lie groups

From Academic Kids

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

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

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

<math>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 \}<math>,

and

<math>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\}<math>,

where <math>*\,\!<math> denotes arbitrary entries in <math>\mathbb{F}_q<math>.

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

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