Somehow, Lie algebras in positive characteristic contain “less information” than in zero characteristic.

Let be a field and a finite-dimensional vector space over . A symplectic form on is a non-degenerate skew-symmetric bilinear form

Here are what the terms mean:

Let be a vector space with a symplectic form. Then there exists a basis of such that

Let

Then with respect to this ordered basis,

where and are column vectors.

The symplectic group is

Then the TheoremTheorem implies that by the Galois cohomology for groups preserving tensors.

We can rewrite the condition as

So we alternatively have the condition . These are quadratic algebraic conditions on the entries of (independent of the field ). So we can repeat this in . In fact, is an algebraic group over .

Over , we write

Now, the condition on is that

This rewrites into

Since the coefficient of must be , the condition on to be in is

This is the type classical Lie algebra.

is almost a simple group. Both , and this is the only normal subgroup of . If is finite, then is a finite simple group.

There exists “similar” (i.e., the construction of , , result is not the same!—there are multiple bilinear forms over an arbitrary field) stories for:

Over , or more generally any which is a separable field extension of degree (automatically Galois), there is a conjugation

which is non-trivial. So we can look at Hermitian forms:

Lie bracket : comes from the non-commutativity of .

Let . We will compute

We use the well known formula

which is a finite sum if is nilpotent. In our example, will be nilpotent since . We have

Therefore

Given , we lift them to . We now consider their commutator