Theorem of the highest weight

In mathematics , the theorem of greatest weight is a fundamental tenet of representation theory that goes back to Elie Cartan . It says that finite-dimensional representations of Lie algebras or Lie groups are uniquely determined by their highest weight.

Terms used

Let be a Lie algebra , a Cartan subalgebra, and a representation . A linear figure ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {h}}}$ ${\displaystyle \pi \colon {\mathfrak {g}}\to {\mathfrak {gl}}(V)}$

${\displaystyle \lambda \colon {\mathfrak {h}}\to \mathbb {C} }$

is called weight of if the weight space ${\displaystyle \pi }$

${\displaystyle V_{\lambda }=\left\{v\in V\colon \pi (h)v=\lambda (h)v\ \forall h\in {\mathfrak {h}}\right\}}$

does not only consist of the zero vector.

The roots of Lie algebra are defined as follows. To be defined by ${\displaystyle R}$ ${\displaystyle \alpha \in {\mathfrak {h}}}$ ${\displaystyle \alpha ^{\vee }\in {\mathfrak {h}}^{*}}$

${\displaystyle \alpha ^{\vee }(x)=2{\frac {B(x,\alpha )}{B(\alpha ,\alpha )}}\ \forall x\in {\mathfrak {h}}}$,

where is the killing form . Then is a root if and only if is a weight of the adjoint representation . ${\displaystyle B}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha ^{\vee }}$ ${\displaystyle ad\colon {\mathfrak {g}}\to {\mathfrak {gl}}(V)}$

After choosing a Weyl chamber , the set of positive roots can be defined by ${\displaystyle {\mathfrak {h}}^{+}}$

${\displaystyle R^{+}:=\left\{\alpha \in R:\alpha ^{\vee }(x)>0\ \forall x\in {\mathfrak {h}}^{+}\right\}}$.

This allows a suborder to be defined on the weights of a given representation

${\displaystyle \lambda \leq \mu \Longleftrightarrow \lambda (\alpha )\leq \mu (\alpha )\ \forall \alpha \in R^{+}}$.

A weight is called the highest weight if there is no greater weight with regard to this partial order.

Furthermore, a linear mapping is called an integral element if ${\displaystyle \lambda \in {\mathfrak {h}}^{*}}$

${\displaystyle \lambda (\alpha )\in \mathbb {Z} \ \ \forall \alpha \in R}$

applies. It is called a dominant integral element when

${\displaystyle \lambda (\alpha )\in \mathbb {N} \ \ \forall \alpha \in R^{+}}$

is.

Theorem of the highest weight

Let be a semi-simple complex Lie algebra. In the following, all representations are finite-dimensional. Then the theorem of highest weight says: ${\displaystyle {\mathfrak {g}}}$

1. Every irreducible representation has a clear highest weight.
2. Two irreducible representations with the same highest weight are equivalent .
3. The greatest weight of an irreducible representation is a dominant integral element.
4. Each dominant integral element carries the greatest weight in an irreducible representation.

Examples

sl (2, C)

A Cartan sub-algebra of is , one can choose as a positive root . For each one has given a dominant integral element through the mapping ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {h}}=\left\{\left({\begin{array}{cc}\lambda &0\\0&-\lambda \end{array}}\right):\lambda \in \mathbb {C} \right\}}$ ${\displaystyle \alpha =\left({\begin{array}{cc}1&0\\0&-1\end{array}}\right)}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \lambda _{n}}$

${\displaystyle \lambda _{n}(\alpha )=n}$.

This corresponds to the known -dimensional irreducible representation (see representation theory of sl (2, C) ) as , where denotes the defining 2-dimensional representation of . ${\displaystyle n}$ ${\displaystyle Sym^{n-1}(V)}$ ${\displaystyle V}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

sl (3, C)

A Cartan subalgebra of is ${\displaystyle {\mathfrak {sl}}(3,\mathbb {C} )}$

${\displaystyle {\mathfrak {h}}=\left\{\left({\begin{array}{ccc}\lambda _{1}&0&0\\0&\lambda _{2}&0\\0&0&\lambda _{3}\end{array}}\right):\lambda _{1},\lambda _{2},\lambda _{3}\in \mathbb {C} \colon \lambda _{1}+\lambda _{2}+\lambda _{3}=0\right\}}$,

as positive roots you can choose and . For each pair one has given a dominant integral element through the mapping ${\displaystyle \alpha _{1}=\left({\begin{array}{ccc}1&0&0\\0&-1&0\\0&0&0\end{array}}\right)}$ ${\displaystyle \alpha _{2}=\left({\begin{array}{ccc}0&0&0\\0&1&0\\0&0&-1\end{array}}\right)}$ ${\displaystyle (m,n)\in \mathbb {N} \times \mathbb {N} }$ ${\displaystyle \lambda _{m,n}}$

${\displaystyle \lambda _{m,n}(\alpha _{1})=m,\lambda _{m,n}(\alpha _{2})=n}$.

The associated representation is a sub-representation of , wherein the defining 3-dimensional representation denotes. More precisely agrees with for the through ${\displaystyle \pi _{m,n}}$ ${\displaystyle Sym^{m}(V)\otimes Sym^{n}(V^{*})}$ ${\displaystyle V}$ ${\displaystyle {\mathfrak {sl}}(3,\mathbb {C} )}$ ${\displaystyle \pi _{m,n}}$ ${\displaystyle Ker(\iota _{m,n})}$

${\displaystyle \iota _{m,n}(v_{1}\ldots v_{m}\otimes v_{1}^{*}\ldots v_{n}^{*})=\sum _{i,j}v_{j}^{*}(v_{i})v_{1}\ldots {\widehat {v_{i}}}\ldots v_{m}\otimes v_{1}^{*}\ldots {\widehat {v_{j}^{*}}}\ldots v_{n}^{*}}$

defined contraction .

Representations of Lie groups

Each representation of a Lie group can be assigned a representation of its Lie algebra , see Representation (Lie algebra) # Representations induced by Lie group representations . In particular, you can define a maximum weight for representations of Lie groups.

Irreducible, finite-dimensional representations of a compact, connected (not necessarily semi-simple) Lie group are classified by their highest weight. This fact is also often referred to as the principle of greatest weight .

literature

• Brian Hall: Lie groups, Lie algebras, and representations. An elementary introduction. Second edition. Graduate Texts in Mathematics, 222. Springer, Cham 2015. ISBN 978-3-319-13466-6