# Witt algebra

Let $k$ be a field of characteristic $p \neq 0$. Consider the $k$- algebra

$$A _ {n} = k [ X _ {1} \dots X _ {n} ] / ( X _ {1} ^ {p} \dots X _ {n} ^ {p} ) .$$

Let $V _ {n}$ be the algebra of $k$- derivations of $A _ {n}$. The algebra $V _ {1}$ is known as the Witt algebra. The $V _ {n}$( $n \geq 2$) are known as the split Jacobson–Witt algebras. The algebra $V _ {n}$ is a simple Lie algebra, except when it is $2$- dimensional. The dimension of $V _ {n}$ is $np ^ {n}$.

More generally one considers the $k$- algebras

$$A _ {n} ( \xi ) = k [ X _ {1} \dots X _ {n} ] / ( X _ {1} ^ {p} - \xi _ {1} \dots X _ {n} ^ {p} - \xi _ {n} ) ,$$

and their algebras of derivations $V _ {n} ( \xi )$, the Jacobson–Witt algebras. The $A _ {n} ( \xi )$ and $V _ {n} ( \xi )$ are (obviously) $k ^ \prime / k$- forms of $A _ {n}$ and $V _ {n}$, where $k ^ \prime = k ( \xi _ {1} ^ {1/p} \dots \xi _ {n} ^ {1/p} )$( cf. Form of an (algebraic) structure). Many simple Lie algebras in characteristic $p$ arise as subalgebras of the $V _ {n}$.

Let $G$ be an additive group of functions on $\{ 1 \dots m \}$ into $k$ such that the only element $f$ of $G$ such that $\sum f( i) g( i) = 0$ for all $g \in G$ is the zero element $f = 0$. For instance, $G$ can be the set of all functions from $\{ 1 \dots m \}$ to some additive subgroup of $k$. If $G$ is finite, it is of order $p ^ {n}$ for some $n$. Now, let $V$ be a vector space over $k$ with basis elements $e _ {g} ^ {i}$, $i = 1 \dots m$, $g \in G$, and define a bilinear product on $V$ by

$$[ e _ {g} ^ {i} , e _ {h} ^ {j} ] = \ h( i) e _ {g+ h } ^ {j} - g( j) e _ {g+ h } ^ {i} .$$

There results a Lie algebra, called a generalized Witt algebra. If $G$ is finite of order $p ^ {n}$, the dimension of $V$ is $m p ^ {n}$, and $V$ is a simple Lie algebra if $m > 1$ or $p > 2$.

If $k$ is of characteristic zero, $m = 1$ and $G$ is the additive subgroup $\mathbf Z \subset k$, the same construction results in the Virasoro algebra $[ e _ {g} , e _ {h} ] = ( h- g) e _ {g+} h$.

If $k$ is of characteristic $p$ and $G$ is the group of all functions on $\{ 1 \dots n \}$ with values in $\mathbf Z / ( p) \subset k$, one recovers the Jacobson–Witt algebras $V _ {n}$.

There are no isomorphisms between the Jacobson–Witt algebras $V _ {n}$ and the classical Lie algebras in positive characteristic when $\mathop{\rm char} ( k) \neq 2, 3$. Several more classes of simple Lie algebras different from the classical ones and the $V _ {n}$ 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.

How to Cite This Entry:
Witt algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Witt_algebra&oldid=49231