# Witt algebra

Jump to: navigation, search

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.