shows the weights of a Verma module for . A simple re-ordering argument shows that there is only one possible way the full Lie algebra can act on this space. Specifically, if is any element of , then by the easy part of the Poincaré–Birkhoff–Witt theorem, we can rewrite as a linear combination of products of Lie algebra elements with the raising operators acting first, the elements of the Cartan subalgebra, and last the lowering operators . Applying this sum of terms to , any term with a raising operator is zero, any factors in the Cartan act as scalars, and thus we end up with an element of the original form. To understand the structure of the Verma module a bit better, we may choose an ordering of the positive roots as and we let the corresponding lowering operators by . Then by a simple re-ordering argument, every element of the above form can be rewritten as a linear combination of elements with the 's in a specific order: , where the 's are non-negative integers. Actually, it turns out that such vectors form a basis for the Verma module. Although this description of the Verma module gives an intuitive idea of what looks like, it still remains to give a rigorous construction of it. In any case, the Verma module gives—for any , not necessarily dominant or integral—a representation with highest weight . The price we pay for this relatively simple construction is that is always infinite dimensional. In the case where is dominant and integral, one can construct a finite-dimensional, irreducible quotient of the Verma module. The case of sl(2; C) Let be the usual basis for : with the Cartan subalgebra being the span of . Let be defined by for an arbitrary complex number . Then the Verma module with highest weight is spanned by linearly independent vectors and the action of the basis elements is as follows: . (This means in particular that and that .) These formulas are motivated by the way the basis elements act in the finite-dimensional representations of , except that we no longer require that the "chain" of eigenvectors for has to terminate. In this construction, is an arbitrary complex number, not necessarily real or positive or an integer. Nevertheless, the case where is a non-negative integer is special. In that case, the span of the vectors is easily