The Submodule of Invariants
If is a module of a Lie algebra , there is one submodule that turns out to be rather interesting: the submodule of vectors such that for all . We call these vectors “invariants” of .
As an illustration of how interesting these are, consider the modules we looked at last time. What are the invariant linear maps from one module to another ? We consider the action of on a linear map :
Or, in other words:
That is, a linear map is invariant if and only if it intertwines the actions on and . That is, .
Next, consider the bilinear forms on . Here we calculate
That is, a bilinear form is invariant if and only if it is associative, in the sense that the Killing form is: