Representations of Lie groups

Abstract

With ξ = (ξ₁,…,ξ _d ), {ξ _i } a basis for the Lie algebra ℊ, a basis for the universal enveloping algebra u(ℊ) is given by

$$\left[\kern-0.15em\left[ \text{n} \right]\kern-0.15em\right] = \xi ^n = \xi _1^{n1} \cdots \xi _d^{nd}$$

where the product is ordered, since the ξ _i do not commute in general. As we saw for Appell systems, it is natural to look at generating functions to see how multiplication by the basis elements ξ _i on u looks. We have

$$\sum\limits_{n \geqslant 0} {\frac{{A^n }} {{n!}}} \,\xi ^n = \sum {\frac{{\left( {A_1 \xi _1 } \right)^{n_1 } }} {{n_1 !}} \ldots \frac{{\left( {A_d \xi _d } \right)^{n_d } }} {{n_d !}} = } \,e^{A_1 \xi _1 } \ldots e^{A_d \xi _d }$$

(0.1)

This is an element of the group, as it is a product of the one-parameter subgroups generated by the basis elements.