Basic Representation Theory

Abstract

If V is a finite-dimensional real or complex vector space, let GL(V ) denote the group of invertible linear transformations of V. If we choose a basis for V, we can identify GL(V ) with $\mathsf{GL}(n; \mathbb{R})$ or $\mathsf{GL}(n; \mathbb{C})$. Any such identification gives rise to a topology on GL(V ), which is easily seen to be independent of the choice of basis. With this discussion in mind, we think of GL(V ) as a matrix Lie group. Similarly, we let gl(V ) = End(V ) denote the space of all linear operators from V to itself, which forms a Lie algebra under the bracket $[X,Y ] = \mathit{XY } -\mathit{YX}$.