Finite-dimensional representations and dominant weights

Let hᵢ be the simple coroots and let fundamental weights ωᵢ satisfy ωᵢ(hⱼ)=δᵢⱼ. A weight is integral when its values on all simple coroots are integers, and dominant when those integers are nonnegative. Thus λ=Σmᵢωᵢ with mᵢ in the nonnegative integers. Restricting a finite highest-weight module to each simple sl(2) triple explains the necessity: its highest-vector string length in that direction is mᵢ+1. Dominance is a lattice condition together with a chamber condition, not merely an arbitrary positive coordinate choice.

The highest-weight classification states that each dominant integral λ produces one finite-dimensional simple L(λ), and every finite-dimensional simple module arises this way, up to isomorphism. The sufficiency direction is a structural theorem requiring more than separately writing candidate rank-one strings; their interactions must satisfy the full algebra's relations. The zero weight labels the trivial one-dimensional module. Changing the positive-root choice changes the selected highest label by the appropriate Weyl convention while preserving the isomorphism class of the underlying representation.

Weyl's complete reducibility theorem says every finite-dimensional complex representation of a semisimple Lie algebra is a direct sum of simple representations. It permits multiplicities: two copies of the same simple module remain two copies. A character records those multiplicities through its weight spaces. Infinite-dimensional Verma modules are outside this theorem, so their non-split submodule structures are not counterexamples. Likewise a Lie algebra with a nontrivial solvable radical need not satisfy the theorem in this form.

Tensor products of finite modules remain finite and decompose into simples, but their highest weights and multiplicities require actual representation-theoretic calculations. The tensor product of highest vectors gives one highest weight equal to the sum, yet it need not generate the whole tensor product. Symmetric and exterior powers provide useful explicit models. When writing polynomials to model a symmetric power, specify whether the variables stand for basis vectors or coordinate functions: the latter are dual and can reverse the expected highest-weight convention.

A finite-dimensional representation integrates to the simply connected complex Lie group with the given semisimple Lie algebra. To obtain a representation of a quotient group, the quotient kernel must act trivially. Thus the algebra's full weight lattice may be larger than the character lattice allowed by a selected global group. For sl(2,C), the highest-weight-m module integrates to SL(2,C), while the central element −I acts by (−1)ᵐ. Exactly the even-m modules descend to PSL(2,C).

The group distinction is not visible in the Lie bracket alone. It explains why an allowed algebraic highest weight can fail to define a representation of a requested non-simply-connected group. A compact real form supplies a positive invariant Hermitian structure on finite modules after group averaging, but that metric is extra representation structure rather than the complex Killing form. Weight-space dimension, vector norm, and measurement probability remain separate notions. A normalized superposition changes a state inside a module, not its multiplicities or its highest-weight label.