Zassenhaus algebra

A Lie algebra of special derivations of the divided power algebra

\begin{equation*} O _ { 1 } ( m ) = \left\{ x ^ { ( i ) } : x ^ { ( i ) } x ^ { ( j ) } = \left( \begin{array} { c } { i + j } \\ { i } \end{array} \right) x ^ { ( i + j ) } , 0 \leq i , j < p ^ { m } \right\} \end{equation*}

over a field $K$ of characteristic $p > 0$. It is usually denoted by $W _ { 1 } ( m )$, is $p ^ { m }$-dimensional and has a basis $\{ e _ { i } : - 1 \leq i \leq p ^ { m } - 2 \}$ with commutator

\begin{equation*} [ e _ { i } , e _ { j } ] = \left( \left( \begin{array} { c } { i + j + 1 } \\ { j } \end{array} \right) - \left( \begin{array} { c } { i + j + 1 } \\ { i } \end{array} \right) \right) e _ { i + j }. \end{equation*}

It also has another basis, $\{ f _ { \alpha } : \alpha \in \operatorname {GF} ( m ) \}$ (if $\operatorname{GF} ( m ) \subseteq K$), with commutator $[ f _ { \alpha } , f _ { \beta } ] = ( \beta - \alpha ) f _ { \alpha + \beta }$, where $\operatorname {GF} ( m )$ is a finite field of order $p ^ { m }$. Zassenhaus algebras appeared first in this form in 1939 [a8] (see also Witt algebra). $W _ { 1 } ( m )$ is simple if $p > 2$ (cf. also Simple algebra), has an ideal of codimension $1$ if $p = 2$, $m > 1$, and is $2$-dimensional non-Abelian if $p = 2$, $m = 1$. It is a Lie $p$-algebra if and only if $m = 1$. The $p$-structure on $W _ { 1 } ( 1 )$ can be given by $ e _{0} ^ { [ p ] } - e _ { 0 } = 0$, $e _ { i } ^ { p } = 0$, $i \neq 0$. By changing $\operatorname {GF} ( m )$ to other additive subgroup of $K$, or by changing the multiplication, one can get different algebras. For example, the multiplication

\begin{equation*} ( f _ { \alpha } , f _ { \beta } ) \mapsto ( \beta - \alpha + h ( \alpha ) \beta - h ( \beta ) \alpha ) f _ { \alpha + \beta }, \end{equation*}

where $h$ is an additive homomorphism of finite fields, gives rise to the Albert–Zassenhaus algebra.

Suppose that all algebras and modules are finite-dimensional and that the ground field $K$ is an algebraically closed field of characteristic $p > 3$.

Let $U _ { t }$ be the $W _ { 1 } ( m )$-module defined for $t \in K$ on the vector space $U = O _ { 1 } ( m )$ by

\begin{equation*} ( e _ { i } ) _ { t } x ^ { ( j ) } = \left( \left( \begin{array} { c } { i + j } \\ { i + 1 } \end{array} \right) + t \left( \begin{array} { c } { i + j } \\ { i } \end{array} \right)\right) x ^ { ( i + j ) }. \end{equation*}

For example, $ U _{ - 1}$ and $U _ { 2 }$ are isomorphic to the adjoint and co-adjoint modules, $U _ { 1 }$ has an irreducible submodule $\overline { U } _ { 1 } = \left\{ x ^ { ( i ) } : 0 \leq i < p ^ { m } - 1 \right\}$, and $U _ { t }$ is irreducible if $t \neq 0,1$. Any irreducible restricted $W _ { 1 } ( 1 )$-module is isomorphic to one of the following modules: the $1$-dimensional trivial module $K$; the $( p - 1 )$-dimensional module $\overline { U } _ { 1 }$; or the $p$-dimensional module $U _ { t }$, $t \in \mathbf{Z} / p \mathbf{Z}$, $t \neq 0,1$. The maximal dimension of irreducible $W _ { 1 } ( m )$-modules is $p ^ { ( p ^ { m } - 1 ) / 2 }$, but there may be infinitely many non-isomorphic irreducible modules of given dimension. The minimal dimension of irreducible non-trivial $W _ { 1 } ( m )$-modules is $p ^ { m } - 1$, and any irreducible module of dimension $p ^ { m } - 1$ is isomorphic to $\overline { U } _ { 1 }$. Any irreducible $W _ { 1 } ( 1 )$-module with a non-trivial action of $e _ { p - 2}$ is irreducible as a module over the maximal subalgebra ${\cal L} _ { 0 } = \langle e _ { i } : i \geq 0 \rangle$. In any case, any non-restricted irreducible $W _ { 1 } ( 1 )$-module is induced by some irreducible submodule of ${\cal L}_0$ [a3].

Any simple Lie algebra of dimension $> 3$ with a subalgebra of codimension $1$ is isomorphic to $W _ { 1 } ( m )$ for some $m$ [a4]. Albert–Zassenhaus algebras have subalgebras of codimension $2$. Any automorphism of $W _ { 1 } ( m )$ is induced by an admissible automorphism of $O _ { 1 } ( m ),$ i.e., by an automorphism $\psi : O _ { 1 } ( m ) \rightarrow O _ { 1 } ( m )$ such that $\psi ( x )$ is a linear combination of $x ^ { ( i ) }$, where $i \neq 0$, $p ^ { k }$, $0 < k < m$. There are infinitely many non-conjugate Cartan subalgebras (cf. also Cartan subalgebra) of dimension $p ^ { m - 1 }$ if $m > 2$, and exactly two non-conjugate Cartan subalgebras of dimension $p$ if $m = 2$ [a2]. The algebra of outer derivations of $W _ { 1 } ( m )$ is $( m - 1 )$-dimensional and generated by the derivations $\left\{ \text { ad } e _ { - 1} ^ { p^k } : 0 < k < m \right\}$. The algebra $W _ { 1 } ( m )$ has a $( 3 m - 2 )$-parametric deformation [a6]. A non-split central extension of $W _ { 1 } ( m )$ was constructed first by R. Block in 1968 [a1]. The characteristic-$0$ infinite-dimensional analogue of this extension is well known as the Virasoro algebra. The list of irreducible $W _ { 1 } ( m )$-modules that have non-split extensions is the following: $M = K , \overline { U } _ { 1 } , U _ { - 1 } , U _ { 2 } , U _ { 3 } , U _ { 5 }$. All bilinear invariant forms of $W _ { 1 } ( m )$ are trivial, but it has a generalized Casimir element $c = \operatorname { ad } e _ { - 1 } ^ { p ^ { m } - 1 } ( e _ { p ^ { m } - 2 } ^ { ( p + 1 ) / 2 } )$. The centre of the universal enveloping algebra is generated by the $p$-centre and the generalized Casimir elements [a7].

References

[a1]	R.E. Block, "On the extension of Lie algebras" Canad. J. Math. , 20 (1968) pp. 1439–1450
[a2]	G. Brown, "Cartan subalgebras of Zassenhaus algebras" Canad. J. Math. , 27 : 5 (1975) pp. 1011–1021
[a3]	Ho-Jui Chang, "Über Wittsche Lie-Ringe" Abb. Math. Sem. Univ. Hamburg , 14 (1941) pp. 151–184
[a4]	A.S. Dzhumadil'daev, "Simple Lie algebras with a subalgebra of codimension 1" Russian Math. Surveys , 40 : 1 (1985) pp. 215–216 (In Russian)
[a5]	A.S. Dzhumadil'daev, "Cohomology and nonsplit extensions of modular Lie algebras" Contemp. Math. , 131:2 (1992) pp. 31–43
[a6]	A.S. Dzhumadil'daev, A.I. Kostrikin, "Deformations of the Lie algebra $W _ { 1 } ( m )$" Proc. Steklov Inst. Math. , 148 (1980) pp. 143–158 (In Russian)
[a7]	Y.B. Ermolaev, "On structure of the center of the universal enveloping algebra of a Zassenhaus algebra" Soviet Math. (Iz.VUZ) , 20 (1978) Izv. VUZ Mat. , 12(199) (1978) pp. 46–59
[a8]	H. Zassenhaus, "Über Lie'she Ringe mit Primzahlcharacteristik" Abh. Math. Sem. Univ. Hamburg , 13 (1939) pp. 1–100