I have two representations of a simple (complex or real) finite-dimensional Lie algebra $S$, both given in terms of their structure constants on a given basis.
- the first one is the adjoint representation on $S$ itself, we write it: $s \in S\mapsto A(s) \in GL(S)$
- the second one is on a given vector space $V$, we write it: $s \in S \mapsto B(s) \in GL(V)$
Would you know an algorithm (or an automatic method) to:
- determine if the two representations of $S$ on $S$ and $V$ are isomorphic,
- and if they are, automatically compute an isomorphism of representations between them?
An isomorphism of $S$-representations $L \in GL$ is defined as: $\forall s \in S, L.A(s)=B(s).L$. Could we use the theory on similar matrices for example?
Many thanks in advance.