Theorem of the highest weight

In representation theory, a branch of mathematics, the theorem of the highest weight states that the irreducible representations of semisimple Lie algebras or compact Lie groups are classified by their highest weights: given a simply-connected compact Lie group G with Lie algebra ${\mathfrak {g}}$ , there is a bijection^[1]

\lambda \mapsto [V^{\lambda }]

from the set of integral points on the positive Weyl chamber, called dominant weights, to the set of equivalence classes of irreducible representations of the complexification of ${\mathfrak {g}}$ (or G); $V^{{\lambda }}$ is an irreducible representation with highest weight $\lambda$ .

Statement

Let ${\mathfrak {g}}$ be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra ${\mathfrak {h}}$ . Let

$\Lambda$ = the ${\mathfrak {h}}$ -weight lattice,
$\Lambda _{\mathbb {R} }$ = the real vector space spanned by $\Lambda$ ,
$\Phi ^{+}$ = set of positive roots, $\Phi ^{-}$ = the set of negative roots,
${\mathfrak {g}}={\mathfrak {g}}_{-}\oplus {\mathfrak {h}}\oplus {\mathfrak {g}}_{+}$ where ${\mathfrak {g}}_{-}$ is spanned by the ${\mathfrak {h}}$ -weight vectors of negative weights and ${\mathfrak {g}}_{+}$ those of positive weights,
$C=\{x\in \Lambda _{\mathbb {R} }\mid \langle x,\alpha \rangle \geq 0,\,\alpha \in \Phi ^{+}\}$ , the positive Weyl chamber.

Then the theorem states:

If V is a finite-dimensional irreducible representation of ${\mathfrak {g}}$ , then the space of vectors v in V such that ${\mathfrak {g}}_{+}v=0$ has dimension 1; a non-zero vector that spans this one-dimensional space is called a highest weight vector of V and the weight of such a vector is called the highest weight of V.
For a highest weight vector v of V, V is spanned by vectors obtained by applying elements of ${\mathfrak {g}}_{-}$ to v.
Every highest weight is dominant in the sense that it lies in C.
If two finite-dimensional irreducible representations have the same highest weight, they are equivalent.
Given a dominant weight (i.e., an integral or lattice point of C), there exists a finite-dimensional irreducible representation whose highest weight is the given dominant weight.^[2]

The most difficult part is the last one; the construction of a finite-dimensional irreducible representation.

Proofs

There are at least three proofs:

The theory of Verma modules contains the highest weight theorem. This is the approach taken in the standard textbooks (e.g., Humphreys and Dixmier).
The Borel–Weil–Bott theorem constructs an irreducible representation as the space of global sections of an ample line bundle; the highest weight theorem results as a consequence. (The approach uses a fair bit of algebraic geometry but yields a very quick proof.)
The invariant theoretic approach: one constructs irreducible representations as subrepresentations of a tensor power of the standard representions. This approach is essentially due to H. Weyl and works quite well for classical groups.

Notes

↑ Dixmier, Theorem 7.2.6.
↑ Dixmier, Proposition 7.2.2. (i)

References

Dixmier, Jacques (1996) [1974], Enveloping algebras, Graduate Studies in Mathematics, 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740
Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
Humphreys, James E. (1972a), Introduction to Lie Algebras and Representation Theory, Birkhäuser, ISBN 978-0-387-90053-7 .

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Dixmier, Theorem 7.2.6.

[2] Dixmier, Proposition 7.2.2. (i)