Topology on a space of Lie algebras

In summary, the conversation discusses the search for a topology on the set of Lie algebras of a given finite dimension that is not defined via structure constants. The goal is to maintain algebraic properties such as solvability, center, and dimension. The proposed approaches include using an algebraic invariant to define the space of Lie algebras or considering a Lie algebra with a specified invariant. However, there are concerns about the effectiveness of the Zariski topology in this context. The main issue is that the invariant is not functorial, meaning that isomorphic Lie algebras may not have isomorphic invariants. The conversation also considers the possibility of defining a topology where naturally Zariski closed sets are open.
  • #1
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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?
 
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  • #2
A lie algebra structure on a vector space ##V## is a bilinear map ##V\times V\to V## satisfying some conditions (skew-symmetry and the Jacobi identity). The space of all bilinear maps ##V\times V\to V## is isomorphic to ##V^*\otimes V^*\otimes V##, which has a natural topology and smooth structure (when ##V## does), and I guess you can give the set of Lie algebras the subspace topology?

Does this work for your question?
 
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  • #3
No. As I said, Zariski- and subspace won't probably work. It might have been a bad idea, in which case I'd like to understand why. I want to consider a point set where the points are equidimensional Lie algebras with a given algebraic property, e.g. being solvable without center. Zariski is the natural choice, but it will either be boring or has too many components to investigate something like a path.

Currently I'm thinking about the following constructions:
  • Define an algebraic invariant, which is a matrix Lie algebra and consider its (unknown) Lie group. In this case the invariant would define the space of Lie algebras.
  • Or, the other way around, consider a Lie algebra (with Lie group) such that the invariant determines the point set.
I just wonder if there are other approaches beside Zariski to define analytic structures on algebraically defined vector spaces. I have this invariant and I am curious in how far I can use it to investigate the many non semisimple Lie algebras. The main problem is that it is not functorial. Isomorphic Lie algebras have isomorphic invariants, but it does not pass to ideals and quotients.

One could ask: is there a topology for a space where naturally Zariski closed sets are open? At the moment I can only think of structure constants as variables, but this would destroy the algebraic properties; even dimension would no longer be constant if we vary those!

Edit: The invariant is ##\{\mathfrak{g}\stackrel{\alpha }{\longrightarrow }\mathfrak{g}\, : \,[\alpha X,Y]+[X,\alpha Y]=0\}##. Easy to check that it is a Lie algebra and that it is also a ##\mathfrak{g}## module.
 

Related to Topology on a space of Lie algebras

1. What is topology on a space of Lie algebras?

Topology on a space of Lie algebras is a mathematical concept that studies the properties and relationships of continuous transformations on a space of Lie algebras. It involves the use of topological spaces and continuous maps to analyze the structure and behavior of these algebraic objects.

2. How is topology used in the study of Lie algebras?

Topology is used in the study of Lie algebras to understand the geometric and structural properties of these algebraic objects. By analyzing the continuous transformations and mappings on a space of Lie algebras, topological tools can provide insights into the behavior and properties of these objects.

3. What are the main applications of topology in the study of Lie algebras?

Topology has various applications in the study of Lie algebras, including the classification of Lie algebras, the study of their representations, and the analysis of their cohomology. It also has applications in other areas of mathematics, such as differential geometry and algebraic topology.

4. How does the topology of a space of Lie algebras affect its properties?

The topology of a space of Lie algebras can greatly influence its properties and behavior. For example, the topology can determine the connectedness, compactness, and completeness of the space, which in turn can affect the existence of certain structures and the behavior of continuous maps on the space.

5. What are some open questions and challenges in the study of topology on a space of Lie algebras?

Despite its many applications and successes, there are still many open questions and challenges in the study of topology on a space of Lie algebras. Some of these include the development of new techniques for analyzing the topology of these spaces, the study of their homotopy and homology groups, and the relationship between the topology and other algebraic structures on these spaces.

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