View source for Zassenhaus algebra

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A [[Lie algebra|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 &lt; p ^ { m } \right\} \end{equation*}

over a [[Field|field]] $K$ of characteristic $p &gt; 0$. 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|finite field]] of order $p ^ { m }$. Zassenhaus algebras appeared first in this form in 1939 [[#References|[a8]]] (see also [[Witt algebra|Witt algebra]]). $W _ { 1 } ( m )$ is simple if $p &gt; 2$ (cf. also [[Simple algebra|Simple algebra]]), has an ideal of codimension $1$ if $p = 2$, $m &gt; 1$, and is $2$-dimensional non-Abelian if $p = 2$, $m = 1$. It is a [[Lie p-algebra|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|algebraically closed field]] of characteristic $p &gt; 3$.

Let $U  _ { t }$ be the $W _ { 1 } ( m )$-module defined for $t \in K$ on the [[Vector space|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 $\overline { U } _ { 1 } = \left\{ x ^ { ( i ) } : 0 \leq i &lt; p ^ { m } - 1 \right\}$, 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 ${\cal L}_0$ [[#References|[a3]]].

Any simple Lie algebra of dimension $&gt; 3$ with a subalgebra of codimension $1$ is isomorphic to $W _ { 1 } ( m )$ for some $m$ [[#References|[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 $i \neq 0$, $p ^ { k }$, $0 &lt; k &lt; m$. There are infinitely many non-conjugate Cartan subalgebras (cf. also [[Cartan subalgebra|Cartan subalgebra]]) of dimension $p ^ { m - 1 }$ if $m &gt; 2$, and exactly two non-conjugate Cartan subalgebras of dimension $p$ if $m = 2$ [[#References|[a2]]]. The algebra of outer derivations of $W _ { 1 } ( m )$ is $( m - 1 )$-dimensional and generated by the derivations $\left\{ \text { ad } e _ { - 1} ^ { p^k } : 0 &lt; k &lt; m \right\}$. The algebra $W _ { 1 } ( m )$ has a $( 3 m - 2 )$-parametric deformation [[#References|[a6]]]. A non-split central extension of $W _ { 1 } ( m )$ was constructed first by R. Block in 1968 [[#References|[a1]]]. The characteristic-$0$ infinite-dimensional analogue of this extension is well known as the [[Virasoro algebra|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|Casimir element]] $c = \operatorname { ad } e _ { - 1 } ^ { p ^ { m } - 1 } ( e _ { p ^ { m } - 2 } ^ { ( p + 1 ) / 2 } )$. The centre of the [[Universal enveloping algebra|universal enveloping algebra]] is generated by the $p$-centre and the generalized Casimir elements [[#References|[a7]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  R.E. Block,   "On the extension of Lie algebras"  ''Canad. J. Math.'' , '''20'''  (1968)  pp. 1439–1450</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  G. Brown,   "Cartan subalgebras of Zassenhaus algebras"  ''Canad. J. Math.'' , '''27''' :  5  (1975)  pp. 1011–1021</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  Ho-Jui Chang,   "Über Wittsche Lie-Ringe"  ''Abb. Math. Sem. Univ. Hamburg'' , '''14'''  (1941)  pp. 151–184</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  A.S. Dzhumadil'daev,   "Simple Lie algebras with a subalgebra of codimension 1"  ''Russian Math. Surveys'' , '''40''' :  1  (1985)  pp. 215–216  (In Russian)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A.S. Dzhumadil'daev,   "Cohomology and nonsplit extensions of modular Lie algebras"  ''Contemp. Math.'' , '''131:2'''  (1992)  pp. 31–43</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  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)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  H. Zassenhaus,   "Über Lie'she Ringe mit Primzahlcharacteristik"  ''Abh. Math. Sem. Univ. Hamburg'' , '''13'''  (1939)  pp. 1–100</td></tr></table>