Some questions about Lie algebras

Question

Why the characteristic of a field is different from two in Lie algebras?

What is the reason for Jacobi's identity in the definition of Lie algebras?

There are Lie algebras of infinite dimension?

Some examples about fields with characteristic two

Thanks for your help

Jaocbi holds iff $\mathfrak g \to \mathfrak{gl}(\mathfrak{g}), x \mapsto \text{ad}(x)$ is a morphisms of Lie algebras. — Nicolas Hemelsoet, Nov 09 '17 at 07:47

score 1 · Answer 1 · answered Nov 09 '17 at 16:28

1.) Why the characteristic of a field is different from two in Lie algebras? The characteristic is arbitrary. However, in characteristic $2$, the skew-symmetry $[x,y]=-[y,x]$ is not enough to conclude that $[v,v]=0$ for all $v$. So the axiom of skew-symmetry is replaced by requiring $[v,v]=0$.

2.) What is the reason for Jacobi's identity in the definition of Lie algebras? The reason is that it comes from the Leibniz rule, differentiating from Lie groups.

3.) There are Lie algebras of infinite dimension? Yes, definitely. There are several books on infinite-dimensional Lie algebras, e.g., the book of Kac.

4.) Some examples about fields with characteristic two: Look into a book on abstract algebra, for finite fields of characteristic $2$. For each power of $2$ there is exactly one such field up to isomorphism.

Some questions about Lie algebras

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