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- TL;DR Summary
- Topology that does not depend on structure constants

I wonder if anybody has an idea for a topology on the set of Lie algebras of a given finite dimension which is not defined via the structure constants. This condition is crucial, as I want to keep as many algebraic properties as possible, e.g. solvability, center, dimension. In the best case the Chevalley-Eilenberg complexes of either a certain given or all representations would be invariant.

The natural question which arises here, is: What is a Lie algebra, if not the set of structure constants? The algebraic properties which I want to keep are all linear functions of the structure constants, so it would be a linear subspace in the end, or at least an affine variety. The topologies that come to mind (Zariski, subspace) are not suited because - I suspect - the affine varieties will be a union of points (irreducible components) in these, and I am looking for a reasonable concept of continuity.

For short: Can you think of a topology (maybe sheaves?) such that a continuous (or even smooth) path in e.g. the set of all five dimensional, slovable, center-less Lie algebras makes sense?

The natural question which arises here, is: What is a Lie algebra, if not the set of structure constants? The algebraic properties which I want to keep are all linear functions of the structure constants, so it would be a linear subspace in the end, or at least an affine variety. The topologies that come to mind (Zariski, subspace) are not suited because - I suspect - the affine varieties will be a union of points (irreducible components) in these, and I am looking for a reasonable concept of continuity.

For short: Can you think of a topology (maybe sheaves?) such that a continuous (or even smooth) path in e.g. the set of all five dimensional, slovable, center-less Lie algebras makes sense?

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