highestweight

Highest weight is a central concept in the representation theory of semisimple Lie algebras and related algebraic structures. It refers to the dominant weight that labels a finite-dimensional irreducible representation. Weights are linear functionals on a Cartan subalgebra, describing how the Cartan elements act by scalars. The highest weight is the maximal element with respect to the dominance order determined by the simple roots.

In a complex semisimple Lie algebra g with Cartan subalgebra h, a weight lambda is called dominant integral if its pairing with each simple coroot α_i∨ is a nonnegative integer. For every such lambda, there exists a unique irreducible finite-dimensional module V(lambda) whose highest weight is lambda. Within V(lambda) there is a highest weight vector v_lambda, annihilated by all positive root spaces, on which h acts by lambda. The remaining weight spaces are obtained by applying lowering operators corresponding to negative roots.

The highest weight determines the representation up to isomorphism: every finite-dimensional irreducible representation has a highest weight, and each dominant integral weight yields one such representation. The set of highest weights forms the lattice of dominant integral weights, and further structure—such as the Weyl character formula and the Weyl dimension formula—allows computation of dimensions and characters. For example, the irreducible sl2(C) module with highest weight n has dimension n+1.

Beyond semisimple Lie algebras, the notion extends to algebraic and quantum groups, where highest weights classify irreducibles and guide constructions like Verma modules and weight multiplicity diagrams. In broader contexts, the term can also denote the largest weight in a weighted collection, but in Lie theory it specifically refers to the dominant label of a representation.