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 , there is a bijection
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 (or G); is an irreducible representation with highest weight .
Let be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra . Let
- = the -weight lattice,
- = the real vector space spanned by ,
- = set of positive roots, = the set of negative roots,
- where is spanned by the -weight vectors of negative weights and those of positive weights,
- , the positive Weyl chamber.
Then the theorem states:
- If V is a finite-dimensional irreducible representation of , then the space of vectors v in V such that 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 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.
The most difficult part is the last one; the construction of a finite-dimensional irreducible representation.
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.
- Dixmier, Jacques (1996) , 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.