Let \mathfrak g be a complex
semisimple Lie algebra and \mathfrak h a
Cartan subalgebra of \mathfrak g. In this section, we describe the concepts needed to formulate the "theorem of the highest weight" classifying the finite-dimensional representations of \mathfrak g. Notably, we will explain the notion of a "dominant integral element." The representations themselves are described in the article linked to above.
Weight of a representation Let \sigma : \mathfrak{g} \to \operatorname{End}(V) be a representation of a Lie algebra \mathfrak g on a vector space
V over a field of characteristic 0, say \mathbb{C}, and let \lambda : \mathfrak{h} \to \mathbb{C} be a linear functional on \mathfrak h, where \mathfrak h is a
Cartan subalgebra of \mathfrak g. Then the ''''
of V
with weight λ'' is the subspace V_\lambda given by :V_\lambda:=\{v\in V: \forall H\in \mathfrak{h},\, (\sigma(H))(v)=\lambda(H)v\}. A
weight of the representation
V (the representation is often referred to in short by the vector space
V over which elements of the Lie algebra act rather than the map \sigma) is a linear functional λ such that the corresponding weight space is nonzero. Nonzero elements of the weight space are called
weight vectors. That is to say, a weight vector is a simultaneous eigenvector for the action of the elements of \mathfrak h, with the corresponding eigenvalues given by λ. If
V is the direct sum of its weight spaces :V=\bigoplus_{\lambda\in\mathfrak{h}^*} V_\lambda then
V is called a
; this corresponds to there being a common
eigenbasis (a basis of simultaneous eigenvectors) for all the represented elements of the algebra, i.e., to there being simultaneously diagonalizable matrices (see
diagonalizable matrix). If
G is a Lie group with Lie algebra \mathfrak g, every finite-dimensional representation of
G induces a representation of \mathfrak g. A weight of the representation of
G is then simply a weight of the associated representation of \mathfrak g. There is a subtle distinction between weights of group representations and Lie algebra representations, which is that there is a different notion of integrality condition in the two cases; see below. (The integrality condition is more restrictive in the group case, reflecting that not every representation of the Lie algebra comes from a representation of the group.)
Action of the root vectors For the
adjoint representation \mathrm{ad} : \mathfrak{g}\to \operatorname{End}(\mathfrak{g}) of \mathfrak g, the space over which the representation acts is the Lie algebra itself. Then the nonzero weights are called
roots, the weight spaces are called
root spaces, and the weight vectors, which are thus elements of \mathfrak{g}, are called
root vectors. Explicitly, a linear functional \alpha on the Cartan subalgebra \mathfrak h is called a root if \alpha\neq 0 and there exists a nonzero X in \mathfrak g such that :[H,X]=\alpha(H)X for all H in \mathfrak h. The collection of roots forms a
root system. From the perspective of representation theory, the significance of the roots and root vectors is the following elementary but important result: If \sigma : \mathfrak{g} \to \operatorname{End}(V) is a representation of \mathfrak g,
v is a weight vector with weight \lambda and
X is a root vector with root \alpha, then : \sigma(H)(\sigma(X)(v))=[(\lambda+\alpha)(H)](\sigma(X)(v)) for all
H in \mathfrak h. That is, \sigma(X)(v) is either the zero vector or a weight vector with weight \lambda+\alpha. Thus, the action of X maps the weight space with weight \lambda into the weight space with weight \lambda+\alpha. For example, if \mathfrak{g}=\mathfrak{su}_{\mathbb{C}}(2), or \mathfrak{su}(2) complexified, the root vectors {H,X,Y} span the algebra and have weights 0, 1, and -1 respectively. The Cartan subalgebra is spanned by H, and the action of H classifies the weight spaces. The action of X maps a weight space of weight \lambda to the weight space of weight \lambda+1 and the action of Y maps a weight space of weight \lambda to the weight space of weight \lambda-1, and the action of H maps the weight spaces to themselves. In the
fundamental representation, with weights \pm\frac{1}{2} and weight spaces V_{\pm\frac{1}{2}}, X maps V_{+\frac{1}{2}} to zero and V_{-\frac{1}{2}} to V_{+\frac{1}{2}}, while Y maps V_{-\frac{1}{2}} to zero and V_{+\frac{1}{2}} to V_{-\frac{1}{2}}, and H maps each weight space to itself.
Integral element Let \mathfrak h^*_0 be the real subspace of \mathfrak h^* generated by the roots of \mathfrak g, where \mathfrak h^* is the space of linear functionals \lambda : \mathfrak h \to \mathbb C, the dual space to \mathfrak h. For computations, it is convenient to choose an inner product that is invariant under the Weyl group, that is, under reflections about the hyperplanes orthogonal to the roots. We may then use this inner product to identify \mathfrak h^*_0 with a subspace \mathfrak h_0 of \mathfrak h. With this identification, the
coroot associated to a root \alpha is given as :H_\alpha=2\frac{\alpha}{(\alpha,\alpha)} where (\alpha,\beta) denotes the
inner product of vectors \alpha,\beta. In addition to this inner product, it is common for an angle bracket notation \langle\cdot,\cdot\rangle to be used in discussions of
root systems, with the angle bracket defined as \langle\lambda,\alpha\rangle\equiv(\lambda,H_\alpha). The angle bracket here is not an inner product, as it is not symmetric, and is linear only in the first argument. The angle bracket notation should not be confused with the inner product (\cdot,\cdot). We now define two different notions of integrality for elements of \mathfrak h_0. The motivation for these definitions is simple: The weights of finite-dimensional representations of \mathfrak g satisfy the first integrality condition, while if
G is a group with Lie algebra \mathfrak g, the weights of finite-dimensional representations of
G satisfy the second integrality condition. An element \lambda\in\mathfrak h_0 is
algebraically integral if :(\lambda,H_\alpha)=2\frac{(\lambda,\alpha)}{(\alpha,\alpha)}\in\mathbb{Z} for all roots \alpha. The motivation for this condition is that the coroot H_\alpha can be identified with the
H element in a standard {X,Y,H} basis for an sl(2,\mathbb C)-subalgebra of \mathfrak g. By elementary results for sl(2,\mathbb C), the eigenvalues of H_\alpha in any finite-dimensional representation must be an integer. We conclude that, as stated above, the weight of any finite-dimensional representation of \mathfrak g is algebraically integral. The
fundamental weights \omega_1,\ldots,\omega_n are defined by the property that they form a basis of \mathfrak h_0 dual to the set of coroots associated to the
simple roots. That is, the fundamental weights are defined by the condition :2\frac{(\omega_i,\alpha_j)}{(\alpha_j,\alpha_j)}=\delta_{i,j} where \alpha_1,\ldots\alpha_n are the simple roots. An element \lambda is then algebraically integral
if and only if it is an integral combination of the fundamental weights. The set of all \mathfrak g-integral weights is a
lattice in \mathfrak h_0 called the
weight lattice for \mathfrak g, denoted by P(\mathfrak g). The figure shows the example of the Lie algebra sl(3,\mathbb C), whose root system is the A_2 root system. There are two simple roots, \gamma_1 and \gamma_2. The first fundamental weight, \omega_1, should be orthogonal to \gamma_2 and should project orthogonally to half of \gamma_1, and similarly for \omega_2. The weight lattice is then the triangular lattice. Suppose now that the Lie algebra \mathfrak g is the Lie algebra of a Lie group
G. Then we say that \lambda\in\mathfrak h_0 is
analytically integral (
G-integral) if for each
t in \mathfrak h such that \exp(t)=1\in G we have (\lambda,t)\in 2\pi i \mathbb{Z}. The reason for making this definition is that if a representation of \mathfrak g arises from a representation of
G, then the weights of the representation will be
G-integral. For
G semisimple, the set of all
G-integral weights is a sublattice
P(
G) ⊂
P(\mathfrak g). If
G is
simply connected, then
P(
G) =
P(\mathfrak g). If
G is not simply connected, then the lattice
P(
G) is smaller than
P(\mathfrak g) and their
quotient is isomorphic to the
fundamental group of
G.
Partial ordering on the space of weights We now introduce a partial ordering on the set of weights, which will be used to formulate the theorem of the highest weight describing the representations of \mathfrak g. Recall that
R is the set of roots; we now fix a set R^+ of
positive roots. Consider two elements \mu and \lambda of \mathfrak h_0. We are mainly interested in the case where \mu and \lambda are integral, but this assumption is not necessary to the definition we are about to introduce. We then say that \mu is
higher than \lambda, which we write as \mu\succeq\lambda, if \mu-\lambda is expressible as a
linear combination of positive roots with non-negative real coefficients. This means, roughly, that "higher" means in the directions of the positive roots. We equivalently say that \lambda is "lower" than \mu, which we write as \lambda\preceq\mu. This is only a
partial ordering; it can easily happen that \mu is neither higher nor lower than \lambda.
Dominant weight An integral element \lambda is
dominant if (\lambda,\gamma)\geq 0 for each positive root \gamma. Equivalently, \lambda is dominant if it is a
non-negative integer combination of the fundamental weights. In the A_2 case, the dominant integral elements live in a 60-degree sector. The notion of being dominant is not the same as being higher than zero. Note the grey area in the picture on the right is a 120-degree sector, strictly containing the 60-degree sector corresponding to the dominant integral elements. The set of all λ (not necessarily integral) such that (\lambda,\gamma)\geq 0 for all positive roots \gamma is known as the
fundamental Weyl chamber associated to the given set of positive roots.
Theorem of the highest weight A weight \lambda of a representation V of \mathfrak g is called a
highest weight if every other weight of V is lower than \lambda. The theory
classifying the finite-dimensional irreducible representations of \mathfrak g is by means of a "theorem of the highest weight." The theorem says that :(1) every irreducible (finite-dimensional) representation has a highest weight, :(2) the highest weight is always a dominant, algebraically integral element, :(3) two irreducible representations with the same highest weight are isomorphic, and :(4) every dominant, algebraically integral element is the highest weight of an irreducible representation. The last point is the most difficult one; the representations may be constructed using
Verma modules.
Highest-weight module A representation (not necessarily finite dimensional)
V of \mathfrak g is called
highest-weight module if it is generated by a weight vector
v ∈
V that is annihilated by the action of all
positive root spaces in \mathfrak g. Every irreducible \mathfrak g-module with a highest weight is necessarily a highest-weight module, but in the infinite-dimensional case, a highest weight module need not be irreducible. For each \lambda\in\mathfrak h^*—not necessarily dominant or integral—there exists a unique (up to isomorphism)
simple highest-weight \mathfrak g-module with highest weight λ, which is denoted
L(λ), but this module is infinite dimensional unless λ is dominant integral. It can be shown that each highest weight module with highest weight λ is a
quotient of the
Verma module M(λ). This is just a restatement of
universality property in the definition of a Verma module. Every
finite-dimensional highest weight module is irreducible. ==See also==