Is a Mapping Between Lie Algebras an Isomorphism if it Takes a Basis to a Basis?

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Homework Help Overview

The discussion revolves around the properties of mappings between Lie algebras, specifically whether a mapping that takes a basis from one Lie algebra to a basis in another can be classified as an isomorphism of vector spaces.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of a mapping that takes a basis to a basis, questioning if this guarantees isomorphism. Some express confidence in the assertion while others seek clarification on the definitions involved.

Discussion Status

The discussion is ongoing, with participants offering varying perspectives on the relationship between basis mappings and isomorphisms. There is an exploration of the assumptions underlying the definitions of Lie algebras and isomorphisms.

Contextual Notes

There is a focus on the definitions of "one to one and onto" in the context of Lie algebras, and how these relate to the concept of a basis.

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If a mapping between Lie algebras [itex]\varphi : \mathfrak{g} \to \mathfrak{h}[/itex] takes a basis in [itex]\mathfrak{g}[/itex] to a basis in [itex]\mathfrak{h}[/itex] is it an isomorphism of vector spaces?
 
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Good question. What do you think?
 


Office_Shredder said:
Good question. What do you think?

I'm fairly sure it is. Is that right?
 


Assuming that by "takes a basis to a basis" you mean "one to one and onto", a Lie Algebra is completely determined by its basis, isn't it?
 

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