Why is it so important for the characteristic value of the field of a lie algebra to not be two for many propositions?

Question

In reading my Lie algebra text, I see a lot of propositions starting with, "If char F does not equal 2, then..." For example, if char F does not equal 2, then o(n,F) is a subalgebra of sl(n,F).

I am failing to see the importance in most cases. Could someone give me an example when char F = 2 will be the downfall? I was thinking about fields where x = -x but couldn't pinpoint the error. I'm sure it's a foolish miss on my part.

Thanks sincerely!

Answer 1

Often times the reason things don't work out in char 2 is because in such fields $2x = 0$ does not imply $x = 0$. This is why for $o(n,F) \subset sl(n,F)$ you need $char F \ne 2$. The Lie algebra $o(n,F)$ is all $A$ such that $A + A^t = 0$. Taking the trace we see that $0 = tr(A + A^t) = tr(A) + tr(A^t) = 2tr(A)$. If $char F \ne 2$ then we conclude that $tr(A) = 0$ so that $A \in sl(n,F)$.

EDIT: actually for $char \ne 2$ any element in $o(n,F)$ must have zero along the diagonal since $A + A^t = 0$ implies that $2a = 0$ for any diagonal entry $a$. In particular the trace must be zero. But for $char = 2$ a skew-symmetric matrix can have an arbitrary diagonal.

So a similar example of where things get messed up in char 2 is that for char not equal to 2, the space of all matrices is the direct sum of anti-symmetric matrices and symmetric matrices. But in char 2 these spaces are the same since $A = A^t$ if and only if $A = -A^t$ since $1 = -1$.

Answer 2

Characteristic three is also problematic for Lie algebras. The appearence of the two characteristics is related to the fact that the identities that define Lie algebras —antisymmetry and the Jacobi identity— are related to the symmetric groups of degree $2$ and $3$, and these have very simple representation theory except when the characteristic is $2$ or $3$.

Answer 3

You hit the nail on the head with thinking "x = -x" because that is equivalent to 2x=0 for all x. I admit openly that I know little about Lie Algebras but I do understand the difficulty of char F = 2.

If char F = 2 the the field F is isomorphic to {0,1} so anytime you appeal to F all it offers is a 0 or a 1 and in that case it is probable your exercise will be rather trivial.

If you are interested in Galois Theory look up Keith Conrad's papers about finding Galois groups via discriminants. He always sets out with char <> 2.