Witt algebra

Let be a field of characteristic . Consider the -algebra

Let be the algebra of -derivations of . The algebra is known as the Witt algebra. The () are known as the split Jacobson–Witt algebras. The algebra is a simple Lie algebra, except when it is -dimensional. The dimension of is .

More generally one considers the -algebras

and their algebras of derivations , the Jacobson–Witt algebras. The and are (obviously) -forms of and , where (cf. Form of an (algebraic) structure). Many simple Lie algebras in characteristic arise as subalgebras of the .

Let be an additive group of functions on into such that the only element of such that for all is the zero element . For instance, can be the set of all functions from to some additive subgroup of . If is finite, it is of order for some . Now, let be a vector space over with basis elements , , , and define a bilinear product on by

There results a Lie algebra, called a generalized Witt algebra. If is finite of order , the dimension of is , and is a simple Lie algebra if or .

If is of characteristic zero, and is the additive subgroup , the same construction results in the Virasoro algebra .

If is of characteristic and is the group of all functions on with values in , one recovers the Jacobson–Witt algebras .

There are no isomorphisms between the Jacobson–Witt algebras and the classical Lie algebras in positive characteristic when . Several more classes of simple Lie algebras different from the classical ones and the are known, [a1].

The Witt algebra(s) described here should of course not be confused with the Witt ring of quadratic forms over a field, nor with the various rings of Witt vectors, cf. Witt vector.

References

[a1]	G.B. Seligman, "Modular Lie algebras" , Springer (1967) MR0245627 Zbl 0189.03201
[a2]	N. Jacobson, "Classes of restricted Lie algebras of characteristic , II" Duke Math. J. , 10 (1943) pp. 107–121
[a3]	R. Ree, "On generalised Witt algebras" Trans. Amer. Math. Soc. , 83 (1956) pp. 510–546