Given Leonid Positselski's excellent answer and comments to this question, I expect that the present one is a hard question. Recall that the Lie algebra structures on a (finite-dimensional over $\mathbb C$, say) vector space $V$ are the maps $\Gamma: V^{\otimes 2} \to V$ satisfying $\Gamma^k_{ij} = -\Gamma^k_{ji}$ and $\Gamma^m_{il}\Gamma^l_{jk} + \Gamma^m_{jl}\Gamma^l_{ki} + \Gamma^m_{kl}\Gamma^l_{ij} = 0$; thus the space of Lie algbera structures is an algebraic variety in $(V^\*)^{\otimes 2} \otimes V$. A Lie algebra structure $\Gamma$ is semisimple if the bilinear pairing $\beta_{ij} = \Gamma^m_{il}\Gamma^l_{jm}$ is nondegenerate; thus the semisimples are a Zariski-open subset of the space of all Lie algebra structures. Because the Cartan classification of isomorphism classes of semisimples is discrete (no continuous families), connected components of the space of semisimples are always contained within isomorphism classes. The semisimples are not dense among all Lie algebra structures: if $\Gamma$ is semisimple, then $\Gamma_{il}^l = 0$, whereas this is not true for the product of the two-dimensional nonabelian with an abelian.
- Is there a (computationally useful) characterization of the Zariski closure of the space of semisimple Lie algebra structures? (LP gives more equations any semisimple satisfies.)
- Suppose that $\Gamma$ is not semisimple but is in the closure of the semisimples. How can I tell for which isomorphism classes of semisimples is $\Gamma$ in the closure of the isomorphism class? (I.e. $\Gamma$ is a limit of what algebras?)
- To what extent can I understand the representation theory of algebras in the closure of the semisimples based on understanding their neighboring semisimple algebras?
For 3., I could imagine the following situation. There is some natural "blow up" of the closure of the semisimples, with at the very least each element of the boundary is a limit of only one isomorphism class in the Cartan classification. Then any representation of the blown-down boundary is some combination of representations of the blown-up parts.
Best Answer
As a warmup to this question you might want to think about the closure of semisimple associative algebras inside all finite dimensional asociative algebras of a given dimension. As a warmup to that question you might want to think about the commutative case. As a warmup to that question you might look at a beautiful paper by Bjorn Poonen (in particular, Section 6 and Remark 6.11).