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) |
[a2] | N. Jacobson, "Classes of restricted Lie algebras of characteristic ![]() |
[a3] | R. Ree, "On generalised Witt algebras" Trans. Amer. Math. Soc. , 83 (1956) pp. 510–546 |
Witt algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Witt_algebra&oldid=17520