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I am trying to learn Lie theory and for this purpose I worked out the Lie algebras of some matrix groups. The examples I worked happened to be complex matrix groups and it lead me to wonder whether, when I try to determine a basis, I had to find a complex or a real basis.

Is there any convention, definition or otherwise a way to know whether a Lie algebra is to be understood as a complex or real vector space?

learner
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2 Answers2

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A real Lie algebra is a real vector space and a complex Lie algebra is a complex vector space. Standard notations for standard Lie algebras usually implicitly specify which is which; for example, $\mathfrak{gl}_n(\mathbb{R})$ is naturally a real Lie algebra while $\mathfrak{gl}_n(\mathbb{C})$ is naturally a complex Lie algebra. If the author does not specify then you need to figure out which is intended from context.

Two warnings: $\mathfrak{u}(n)$ is a real Lie algebra. It does not have a natural complex structure. And people frequently take complexifications $\mathfrak{g} \otimes_{\mathbb{R}} \mathbb{C}$ of real Lie algebras, e.g. when determining their complex representation theory.

Qiaochu Yuan
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  • If it's not specified but the Lie groups are complex, like $SL_n(\mathbb C)$, should I assume that the Lie algebras are taken to be complex? (I was showing that two Lie algebras are isomorphic and it might not matter for this purpose...?) – learner Feb 01 '15 at 23:21
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    @learner: probably, but keep in mind that $U(n)$, for example, is not a complex Lie group. – Qiaochu Yuan Feb 02 '15 at 00:30
  • Could you explain a bit about this notation "$⊗_ℝ ℂ$ "? Is this a tensor product $⊗$ and ℝ is for the ring or for the real number? (I find https://math.stackexchange.com/questions/181788/tensor-product-algebra-mathbbc-otimes-mathbbr-mathbbc) – annie marie cœur May 11 '21 at 17:52
  • Another question: () is a real Lie group. But () matrix representation is usually written in G$$(ℂ) which seems generally a complex Lie group. So how do I digest the difference between () vs G$$(ℂ)? – annie marie cœur May 11 '21 at 17:54
  • Many thanks!!! voted 1 – annie marie cœur May 11 '21 at 17:54
  • Could you comment? – annie marie cœur May 13 '21 at 19:33
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In general a Lie algebra is defined over an arbitrary field. For connection with $p$-groups, the field may be also a finite field $\mathbb{F}_q$ of characteristic $p>0$. Many results for Lie algebras, however, are only true over an algebraically closed field of characteristic zero, e.g., Lie's theorem for Lie algebras. Note that the classification of finite-dimensional simple Lie algebras in characteristic $p>0$ differs drastically from the one in characteristic $0$.

Dietrich Burde
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