Peso (teoría de la representación)

Dejar${\displaystyle \sigma$ :{\mathfrak {g}}\to \operatorname {End} (V)} sea una representación de un álgebra de Lie${\displaystyle {\mathfrak {g}}}$en un espacio vectorial V sobre un cuerpo de característica 0, por ejemplo${\displaystyle \mathbb {C} }$y dejar${\displaystyle \lambda$ :{\mathfrak {h}}\to \mathbb {C} } sea un funcional lineal en${\displaystyle {\mathfrak {h}}}$, dónde${\displaystyle {\mathfrak {h}}}$es una subálgebra de Cartan de${\displaystyle {\mathfrak {g}}}$. Entonces elespacio de peso deVcon pesoλes el subespacio${\displaystyle V_{\lambda }}$dado por

${\displaystyle V_{\lambda }:=\{v\in V:\forall H\in {\mathfrak {h}},\,(\sigma (H))(v)=\lambda (H)v\}}$.

Un peso de la representación V (la representación a menudo se denomina brevemente espacio vectorial V sobre el cual actúan los elementos del álgebra de Lie en lugar del mapa${\displaystyle \sigma }$) es un funcional lineal λ tal que el espacio de pesos correspondiente es distinto de cero. Los elementos distintos de cero del espacio de pesos se denominan vectores de peso . Es decir, un vector de peso es un vector propio simultáneo para la acción de los elementos de${\displaystyle {\mathfrak {h}}}$, con los autovalores correspondientes dados por λ.

Si V es la suma directa de sus espacios de peso

${\displaystyle V=\bigoplus _{\lambda \in {\mathfrak {h}}^{*}}V_{\lambda }}$

entonces V se llama unmódulo de peso ;esto corresponde a que exista unabase de autovectores(una base de autovectores simultáneos) para todos los elementos representados del álgebra, es decir, a que existan matrices diagonalizables simultáneamente (véasematriz diagonalizable).

Si G es un grupo de Lie con álgebra de Lie${\displaystyle {\mathfrak {g}}}$, toda representación de dimensión finita de G induce una representación de${\displaystyle {\mathfrak {g}}}$. Un peso de la representación de G es entonces simplemente un peso de la representación asociada de${\displaystyle {\mathfrak {g}}}$Existe una sutil distinción entre los pesos de las representaciones de grupos y las representaciones de álgebras de Lie, que radica en la diferente noción de condición de integralidad en ambos casos; véase más adelante. (La condición de integralidad es más restrictiva en el caso de los grupos, lo que refleja que no toda representación del álgebra de Lie proviene de una representación del grupo).

Para la representación adjunta${\displaystyle \mathrm {ad}$ :{\mathfrak {g}}\to \operatorname {End} ({\mathfrak {g}})} de ${\displaystyle {\mathfrak {g}}}$, el espacio sobre el que actúa la representación es el álgebra de Lie misma. Entonces los pesos no nulos se llaman raíces , los espacios de pesos se llaman espacios de raíces y los vectores de pesos, que son por lo tanto elementos de${\displaystyle {\mathfrak {g}}}$, se denominan vectores raíz . Explícitamente, una función lineal${\displaystyle \alpha }$sobre la subálgebra de Cartan${\displaystyle {\mathfrak {h}}}$se llama raíz si${\displaystyle \alpha \neq 0}$y existe un valor distinto de cero${\displaystyle X}$en${\displaystyle {\mathfrak {g}}}$de tal manera que

${\displaystyle [H,X]=\alpha (H)X}$

a pesar de${\displaystyle H}$en${\displaystyle {\mathfrak {h}}}$El conjunto de raíces forma un sistema radicular .

Desde la perspectiva de la teoría de la representación, la importancia de las raíces y los vectores raíz radica en el siguiente resultado elemental pero importante: Si${\displaystyle \sigma$ :{\mathfrak {g}}\to \operatorname {End} (V)} es una representación de${\displaystyle {\mathfrak {g}}}$, v es un vector de pesos con peso${\displaystyle \lambda }$y X es un vector raíz con raíz${\displaystyle \alpha }$, entonces

${\displaystyle \sigma (H)(\sigma (X)(v))=[(\lambda +\alpha )(H)](\sigma (X)(v))}$

para todo H en${\displaystyle {\mathfrak {h}}}$. Eso es,${\displaystyle \sigma (X)(v)}$ is either the zero vector or a weight vector with weight ${\displaystyle \lambda +\alpha }$. Thus, the action of ${\displaystyle X}$ maps the weight space with weight ${\displaystyle \lambda }$ into the weight space with weight ${\displaystyle \lambda +\alpha }$.

For example, if ${\displaystyle {\mathfrak {g}}={\mathfrak {su}}_{\mathbb {C} }(2)}$, or ${\displaystyle {\mathfrak {su}}(2)}$ complexified, the root vectors ${\displaystyle {H,X,Y}}$ span the algebra and have weights ${\displaystyle 0}$, ${\displaystyle 1}$, and ${\displaystyle -1}$ respectively. The Cartan subalgebra is spanned by ${\displaystyle H}$, and the action of ${\displaystyle H}$ classifies the weight spaces. The action of ${\displaystyle X}$ maps a weight space of weight ${\displaystyle \lambda }$ to the weight space of weight ${\displaystyle \lambda +1}$ and the action of ${\displaystyle Y}$ maps a weight space of weight ${\displaystyle \lambda }$ to the weight space of weight ${\displaystyle \lambda -1}$, and the action of ${\displaystyle H}$ maps the weight spaces to themselves. In the fundamental representation, with weights ${\displaystyle \pm {\frac {1}{2}}}$ and weight spaces ${\displaystyle V_{\pm {\frac {1}{2}}}}$, ${\displaystyle X}$ maps ${\displaystyle V_{+{\frac {1}{2}}}}$ to zero and ${\displaystyle V_{-{\frac {1}{2}}}}$ to ${\displaystyle V_{+{\frac {1}{2}}}}$, while ${\displaystyle Y}$ maps ${\displaystyle V_{-{\frac {1}{2}}}}$ to zero and ${\displaystyle V_{+{\frac {1}{2}}}}$ to ${\displaystyle V_{-{\frac {1}{2}}}}$, and ${\displaystyle H}$ maps each weight space to itself.

Let ${\displaystyle {\mathfrak {h}}_{0}^{*}}$ be the real subspace of ${\displaystyle {\mathfrak {h}}^{*}}$ generated by the roots of ${\displaystyle {\mathfrak {g}}}$, where ${\displaystyle {\mathfrak {h}}^{*}}$ is the space of linear functionals ${\displaystyle \lambda$ :{\mathfrak {h}}\to \mathbb {C} } , the dual space to ${\displaystyle {\mathfrak {h}}}$. 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 ${\displaystyle {\mathfrak {h}}_{0}^{*}}$ with a subspace ${\displaystyle {\mathfrak {h}}_{0}}$ of ${\displaystyle {\mathfrak {h}}}$. With this identification, the coroot associated to a root ${\displaystyle \alpha }$ is given as

${\displaystyle H_{\alpha }=2{\frac {\alpha }{(\alpha ,\alpha )}}}$

where ${\displaystyle (\alpha ,\beta )}$ denotes the inner product of vectors ${\displaystyle \alpha ,\beta .}$ In addition to this inner product, it is common for an angle bracket notation ${\displaystyle \langle \cdot ,\cdot \rangle }$ to be used in discussions of root systems, with the angle bracket defined as ${\displaystyle \langle \lambda ,\alpha \rangle \equiv (\lambda ,H_{\alpha }).}$ 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 ${\displaystyle (\cdot ,\cdot ).}$

We now define two different notions of integrality for elements of ${\displaystyle {\mathfrak {h}}_{0}}$. The motivation for these definitions is simple: The weights of finite-dimensional representations of ${\displaystyle {\mathfrak {g}}}$ satisfy the first integrality condition, while if G is a group with Lie algebra ${\displaystyle {\mathfrak {g}}}$, the weights of finite-dimensional representations of G satisfy the second integrality condition.

An element ${\displaystyle \lambda \in {\mathfrak {h}}_{0}}$ is algebraically integral if

${\displaystyle (\lambda ,H_{\alpha })=2{\frac {(\lambda ,\alpha )}{(\alpha ,\alpha )}}\in \mathbb {Z} }$

for all roots ${\displaystyle \alpha }$. The motivation for this condition is that the coroot ${\displaystyle H_{\alpha }}$ can be identified with the H element in a standard ${\displaystyle {X,Y,H}}$ basis for an ${\displaystyle sl(2,\mathbb {C} )}$-subalgebra of ${\displaystyle {\mathfrak {g}}}$.^[1] By elementary results for ${\displaystyle sl(2,\mathbb {C} )}$, the eigenvalues of ${\displaystyle H_{\alpha }}$ in any finite-dimensional representation must be an integer. We conclude that, as stated above, the weight of any finite-dimensional representation of ${\displaystyle {\mathfrak {g}}}$ is algebraically integral.^[2]

The fundamental weights${\displaystyle \omega _{1},\ldots ,\omega _{n}}$ are defined by the property that they form a basis of ${\displaystyle {\mathfrak {h}}_{0}}$dual al conjunto de corraíces asociadas a las raíces simples . Es decir, los pesos fundamentales se definen por la condición

${\displaystyle 2{\frac {(\omega _{i},\alpha _{j})}{(\alpha _{j},\alpha _{j})}}=\delta _{i,j}}$

dónde ${\displaystyle \alpha _{1},\ldots \alpha _{n}}$son las raíces simples. Un elemento${\displaystyle \lambda }$es entonces algebraicamente integral si y solo si es una combinación integral de los pesos fundamentales. ^{[ 3 ]} El conjunto de todos${\displaystyle {\mathfrak {g}}}$-pesos integrales es una red en${\displaystyle {\mathfrak {h}}_{0}}$llamada la red de peso para${\displaystyle {\mathfrak {g}}}$, denotado por${\displaystyle P({\mathfrak {g}})}$.

La figura muestra un ejemplo del álgebra de Lie.${\displaystyle sl(3,\mathbb {C} )}$, cuyo sistema radicular es el${\displaystyle A_{2}}$sistema radicular. Hay dos raíces simples,${\displaystyle \gamma _{1}}$y${\displaystyle \gamma _{2}}$. El primer peso fundamental,${\displaystyle \omega _{1}}$, debería ser ortogonal a${\displaystyle \gamma _{2}}$y debería proyectarse ortogonalmente a la mitad de${\displaystyle \gamma _{1}}$y de manera similar para${\displaystyle \omega _{2}}$. La red de pesos es entonces la red triangular.

Supongamos ahora que el álgebra de Lie${\displaystyle {\mathfrak {g}}}$es el álgebra de Lie de un grupo de Lie G. Entonces decimos que${\displaystyle \lambda \in {\mathfrak {h}}_{0}}$es analíticamente integral ( integral G ) si para cada t en${\displaystyle {\mathfrak {h}}}$de tal manera que${\displaystyle \exp(t)=1\in G}$tenemos${\displaystyle (\lambda ,t)\in 2\pi i\mathbb {Z} }$. La razón para hacer esta definición es que si una representación de${\displaystyle {\mathfrak {g}}}$surge de una representación de G , entonces los pesos de la representación serán G -integrales. ^{[ 4 ]} Para G semisimple, el conjunto de todos los pesos G -integrales es una subred P ( G ) ⊂ P (${\displaystyle {\mathfrak {g}}}$). Si G es simplemente conexo , entonces P ( G ) = P (${\displaystyle {\mathfrak {g}}}$). Si G no es simplemente conexo, entonces la red P ( G ) es más pequeña que P (${\displaystyle {\mathfrak {g}}}$) y su cociente es isomorfo al grupo fundamental de G . ^{[ 5 ]}

Ahora introducimos un orden parcial en el conjunto de pesos, que se utilizará para formular el teorema del peso más alto que describe las representaciones de${\displaystyle {\mathfrak {g}}}$Recordemos que R es el conjunto de raíces; ahora fijamos un conjunto${\displaystyle R^{+}}$de raíces positivas .

Consideremos dos elementos${\displaystyle \mu }$y${\displaystyle \lambda }$de ${\displaystyle {\mathfrak {h}}_{0}}$. Nos interesa principalmente el caso en el que${\displaystyle \mu }$y${\displaystyle \lambda }$son integrales, pero esta suposición no es necesaria para la definición que estamos a punto de introducir. Entonces decimos que${\displaystyle \mu }$es más alto que${\displaystyle \lambda }$, que escribimos como${\displaystyle \mu \succeq \lambda }$, si${\displaystyle \mu -\lambda }$ is expressible as a linear combination of positive roots with non-negative real coefficients.^[6] This means, roughly, that "higher" means in the directions of the positive roots. We equivalently say that ${\displaystyle \lambda }$ is "lower" than ${\displaystyle \mu }$, which we write as ${\displaystyle \lambda \preceq \mu }$.

This is only a partial ordering; it can easily happen that ${\displaystyle \mu }$ is neither higher nor lower than ${\displaystyle \lambda }$.

An integral element ${\displaystyle \lambda }$ is dominant if ${\displaystyle (\lambda ,\gamma )\geq 0}$ for each positive root ${\displaystyle \gamma }$. Equivalently, ${\displaystyle \lambda }$ is dominant if it is a non-negative integer combination of the fundamental weights. In the ${\displaystyle A_{2}}$ 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 ${\displaystyle (\lambda ,\gamma )\geq 0}$ for all positive roots ${\displaystyle \gamma }$ is known as the fundamental Weyl chamber associated to the given set of positive roots.

A weight ${\displaystyle \lambda }$ of a representation ${\displaystyle V}$ of ${\displaystyle {\mathfrak {g}}}$ is called a highest weight if every other weight of ${\displaystyle V}$ is lower than ${\displaystyle \lambda }$.

The theory classifying the finite-dimensional irreducible representations of ${\displaystyle {\mathfrak {g}}}$ is by means of a "theorem of the highest weight." The theorem says that^[7]

(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.

A representation (not necessarily finite dimensional) V of ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle {\mathfrak {g}}}$. Every irreducible ${\displaystyle {\mathfrak {g}}}$-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 ${\displaystyle \lambda \in {\mathfrak {h}}^{*}}$—not necessarily dominant or integral—there exists a unique (up to isomorphism) simple highest-weight ${\displaystyle {\mathfrak {g}}}$-módulo con peso máximo λ, que se denota L (λ), pero este módulo es de dimensión infinita a menos que λ sea una integral dominante. Se puede demostrar que cada módulo de peso máximo con peso máximo λ es un cociente del módulo de Verma M (λ). Esto es simplemente una reformulación de la propiedad de universalidad en la definición de un módulo de Verma.

Todo módulo de peso máximo de dimensión finita es irreducible. ^{[ 8 ]}