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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.

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2 Answers 2

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In general, we can write down the polynomial equations for two $S$-modules being isomorphic. By considering characteristic polynomials of similar matrices one can sometimes get necessary conditions which simplify the equations. Also, the non-vanishing of the determinant is useful (the morphism is bijective). If the dimension of $S$ is not too high, then the equations can just be solved by computing a Groebner basis. This works in the same way as testing Lie algebras for isomorphism. The book of Willem de Graaf has a discussion on this.

Of course, if one has additional information in form of invariants which are easy to compute, this will make the computations much easier. Also, the dimensions of the cohomology groups $H^n(L,M)$ are useful invariants. Sometimes however this does not help too much, e.g., the invariants could just be identical.

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  • $\begingroup$ Many thanks! I will delve into that. $\endgroup$
    – Nina
    Commented Jan 21, 2015 at 9:30
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Working over $\mathbb{C}$, the simplicity of $S$ implies that the adjoint representation is irreducible. Picking a Cartan subalgebra of $S$ and a collection of positive roots, it follows that the adjoint representation is determined up to isomorphism by its highest weight (ie. highest root).

So, perhaps you could begin by determining whether your second representation is irreducible. If it is reducible, then your representations cannot be isomorphic. If it is irreducible, then try to find its highest weight. If this agrees with the highest root, then your representations are isomorphic.

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  • $\begingroup$ Yes thanks, I can do that. But, would you also know a generalization to the real Lie algebra case? $\endgroup$
    – Nina
    Commented Jan 20, 2015 at 17:41
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    $\begingroup$ I'm not sure about a complete generalization, but you can tensor your representations with $\mathbb{C}$ to get representations of a complex simple Lie algebra. You can then determine if these complex representations are isomorphic using the above strategy. If not, then then the original real representations cannot be isomorphic. $\endgroup$
    – Peter Crooks
    Commented Jan 20, 2015 at 17:47
  • $\begingroup$ Yes I see, thank you for the suggestion. This may answer part of the issue. However I am looking for a complete characterization of them being isomorphic or not. Therefore: what if the complexified representations are isomorphic using the above strategy? Would you have an idea? Many thanks again. $\endgroup$
    – Nina
    Commented Jan 20, 2015 at 17:56
  • $\begingroup$ I think I would then look for some distinguishing properties of the adjoint representation over the reals. One such property is that the adjoint representation is faithful since your algebra is simple. I would then try to determine whether my other representation was faithful. Overall, however, I cannot think of a more systematic approach. $\endgroup$
    – Peter Crooks
    Commented Jan 20, 2015 at 18:02
  • $\begingroup$ Okay, thank you. I will have a look at this. $\endgroup$
    – Nina
    Commented Jan 20, 2015 at 18:05

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