factor of 2, the elements , and may be identified with the angular momentum operators , , and , respectively. The factor of 2 is a discrepancy between conventions in math and physics; we will attempt to mention both conventions in the results that follow. Weights and the structure of the representation In this setting, the eigenvalues for are referred to as the weights of the representation. The following elementary result is a key step in the analysis. Suppose that is an eigenvector for with eigenvalue that is, that Then In other words, is either the zero vector or an eigenvector for with eigenvalue and is either zero or an eigenvector for with eigenvalue Thus, the operator acts as a raising operator, increasing the weight by 2, while acts as a lowering operator. Suppose now that is an irreducible, finite-dimensional representation of the complexified Lie algebra. Then can have only finitely many eigenvalues. In particular, there must be some final eigenvalue with the property that is not an eigenvalue. Let be an eigenvector for with that eigenvalue then we must have or else the above identity would tell us that is an eigenvector with eigenvalue Now define a "chain" of vectors by . A simple argument by induction then shows that for all Now, if is not the zero vector, it is an eigenvector for with eigenvalue Since, again, has only finitely many eigenvectors, we conclude that must be zero for some (and then for all ). Let be the last nonzero vector in the chain; that is, but Then of course and by the above identity with we have Since is at least one and we conclude that must be equal to the non-negative integer We thus obtain a chain of vectors, such that acts as and acts as and acts as (We have replaced with its currently known value of in the formulas above.) Since the vectors are eigenvectors for with distinct eigenvalues, they must be linearly independent. Furthermore, the span of is clearly invariant under the action of the complexified Lie algebra. Since is assumed irreducible, this span must be all of We thus obtain a complete description of what an irreducible representation must look like; that is, a basis for the space and a complete description of how the generators of the Lie algebra act. Conversely, for any we can construct a representation by simply using