Weight (representation theory)

From Wikipedia, the free encyclopedia
Jump to: navigation, search


Motivation and general concept[edit]




 \chi(gh) = \chi(g)\chi(h) for all g, h in G.

Indeed, if G acts on a vector space V over F, each simultaneous eigenspace for every element of G, if such exists, determines a multiplicative character on G; the eigenvalue on this common eigenspace of each element of the group.



for all a, b in A. If an algebra A acts on a vector space V over F to any simultaneous eigenspace corresponds an algebra homomorphism from A to F assigning to each element of A its eigenvalue.



Weight space of a representation[edit]


V_\lambda:=\{v\in V: \forall \xi\in \mathfrak{g},\quad  \xi\cdot v=\lambda(\xi)v\}

(where \xi \cdot v denotes the action of g on V). A weight of the representation V is a weight λ such that the corresponding weight space is nonzero. Nonzero elements of the weight space are called weight vectors.

If V is the direct sum of its weight spaces

V=\bigoplus_{\lambda\in\mathfrak{g}^*} V_\lambda

then it is called a weight module; this corresponds to having an eigenbasis (a basis of eigenvectors), i.e., being a diagonalizable matrix.


Semisimple Lie algebras[edit]




We now assume that g is semisimple, with a chosen Cartan subalgebra h and corresponding root system. Let us suppose also that a choice of positive roots Φ+ has been fixed. This is equivalent to the choice of a set of simple roots.

Ordering on the space of weights[edit]

Let h*0 be the real subspace of h* (if it is complex) generated by the roots of g.

There are two concepts how to define an ordering of h*0.

The first one is

μ ≤ λ if and only if λ − μ is nonnegative linear combination of simple roots.

The second concept is given by an element f in h0 and

μ ≤ λ if and only if μ(f) ≤ λ(f).

Usually, f is chosen so that β(f) > 0 for each positive root β.

Integral weight[edit]

A weight λ ∈ h* is integral (or g-integral), if λ(Hγ) ∈ Z for each coroot Hγ such that γ is a positive root.

The fundamental weights \omega_1,\ldots,\omega_n are defined by the property that they form a basis of h* dual to the set of simple coroots H_{\alpha_1}, \ldots, H_{\alpha_n}.

Hence λ is integral if it is an integral combination of the fundamental weights. The set of all g-integral weights is a lattice[disambiguation needed] in h* called weight lattice for g, denoted by P(g).

A weight λ of the Lie group G is called integral, if for each t in h such that \exp(t)=1\in G,\,\,\lambda(t)\in 2\pi i \mathbf{Z}. For G semisimple, the set of all G-integral weights is a sublattice P(G) ⊂ P(g). If G is simply connected, then P(G) = P(g). If G is not simply connected, then the lattice P(G) is smaller than P(g) and their quotient is isomorphic to the fundamental group of G.

Dominant weight[edit]

A weight λ is dominant if \lambda(H_\gamma)\geq 0 for each coroot Hγ such that γ is a positive root. Equivalently, λ is dominant, if it is a non-negative linear combination of the fundamental weights.



Highest weight[edit]

A weight λ of a representation V is called the highest weight if no other weight of V is larger than λ. Sometimes, it is assumed that a highest weight is a weight, such that all other weights of V are strictly smaller than λ in the partial ordering given above. The term highest weight denotes often the highest weight of a "highest-weight module".

Similarly, we define the lowest weight.




A weight vector v_\lambda \in V of weight λ is called a highest-weight vector, or vector of highest weight, if all other weights of V are smaller than λ.

Highest-weight module[edit]

A representation V of g is called highest-weight module if it is generated by a weight vector vV that is annihilated by the action of all positive root spaces in g.

This is something more special than a g-module with a highest weight.

Similarly we can define a highest-weight module for representation of a Lie group or an associative algebra.

Verma module[edit]

Main article: Verma module

For each dominant weight λ ∈ h*, there exists a unique (up to isomorphism) simple highest-weight g-module with highest weight λ, which is denoted L(λ).


A highest-weight module is a weight module. The weight spaces in a highest-weight module are always finite dimensional.

See also[edit]

