<Section Label="nonsolvdescr">
<Heading>Description of the non-solvable Lie algebras</Heading>
In this section we list the non-solvable
Lie algebras contained in the package.
Our notation follows <Cite Key="Strade"/>, where a more detailed description
can also be found. In particular if <Math>L</Math> is a Lie algebra over
<Math>F</Math> then <Math>C(L)</Math> denotes the center of <Math>L</Math>. Further, if <Math>x_1,\ldots,x_k</Math> are elements of <Math>L</Math>, then
<Math>F&tlt;x_1,\ldots,x_k&tgt;</Math> denotes the linear subspace generated
by <Math>x_1,\ldots,x_k</Math>, and we also write
<Math>Fx_1</Math> for <Math>F&tlt;x_1&tgt;</Math>
</Section>
There are no non-solvable Lie algebras with dimension 1 or 2. Over an
arbitrary finite field <A>F</A>, there is just one isomorphism type of non-solvable
Lie algebras:
<Enum>
<Item> If <A>char F=2</A> then the algebra is <Math>W(1;\underline 2)^{(1)}</Math>.
</Item>
<Item> If <A>char F>2</A> then the algebra is <Math>\mbox{sl}(2,F)</Math>.</Item>
</Enum>
See Theorem 3.2 of <Cite Key="Strade"/> for details.
</Section>
<Section Label="appdim4">
<Heading>Dimension 4</Heading>
Over a finite field <A>F</A>
of characteristic 2 there are two isomorphism classes of
non-solvable Lie algebras with dimension 4,
while over a finite field <A>F</A> of odd characteristic
the number of isomorphism classes is one (see Theorem 4.1 of <Cite
Key="Strade"/>). The classes are as follows:
<Enum>
<Item> characteristic 2: <Math>W(1;\underline 2)</Math> and
<Math>W(1;\underline 2)^{(1)}\oplus F</Math>. </Item>
<Item> odd characteristic: <Math>\mbox{gl}(2,F)</Math>.</Item>
</Enum>
</Section>
<Subsection Label="appdim5char2">
<Heading>Characteristic 2</Heading>
Over a finite field <A>F</A>
of characteristic 2 there are 5 isomorphism classes of
non-solvable Lie algebras with dimension 5:
<Enum>
<Item> <Math>\mbox{Der}(W(1;\underline 2)^{(1)})</Math>;</Item>
<Item> <Math>W(1;\underline 2)\ltimes Fu</Math> where <Math>[W(1;\underline 2)^{(1)},u]=0</Math>, <Math>[x^{(3)}\partial,u]=\delta u</Math> and <Math>\delta\in\{0,1\}</Math>
(two algebras);</Item>
<Item> <Math>W(1;\underline 2)^{(1)}\oplus(F\left&tlt; h,u\right&tgt;)</Math>, <Math>[h,u]=\delta u</Math>, where <Math>\delta\in\{0,1\}</Math> (two algebras).</Item>
</Enum>
See Theorem 4.2 of <Cite Key="Strade"/> for details.
</Subsection>
<Subsection Label="appdim5charodd"><Heading>Odd characteristic</Heading>
Over a field <Math>F</Math>of odd characteristic the number of
isomorphism types of 5-dimensional
non-solvable Lie algebras is <Math>3</Math> if the characteristic is at least
7, and it is 4 otherwise (see Theorem 4.3 of <Cite Key="Strade"/>).
The classes are as follows.
<Enum>
<Item><Math>\mbox{sl}(2,F)\oplus F&tlt;x,y&tgt;</Math>, <Math>[x,y]=\delta y</Math> where <Math>\delta\in\{0,1\}</Math>.</Item>
<Item><Math>\mbox{sl}(2,F)\ltimes V(1)</Math> where <Math>V(1)</Math> is the irreducible 2-dimensional <Math>\mbox{sl}(2,F)</Math>-module.</Item>
<Item>If <Math>\mbox{char }F=3</Math> then there is an additional algebra, namely
the non-split extension <Math>0\rightarrow V(1)\rightarrow L\rightarrow\mbox{sl}(2,F)\rightarrow 0</Math>.</Item>
<Item>If <Math>\mbox{char }F=5</Math> then there is an additional algebra: <Math>W(1;\underline 1)</Math>.
</Item></Enum>
</Subsection>
</Section>
<Section Label="appdim6"><Heading>Dimension 6</Heading>
<Subsection Label="appdim6char2"><Heading>Characteristic 2</Heading>
Over a field <Math>F</Math> of characteristic 2, the isomorphism
classes of non-solvable Lie algebras are as follows.
<Enum><Item><Math>W(1;\underline 2)^{(1)}\oplus W(1;\underline 2)^{(1)}</Math>.</Item>
<Item><Math>W(1;\underline 2)^{(1)}\otimes F_{q^2}</Math> where <Math>F=F_q</Math>.</Item>
<Item><Math>\mbox{Der}(W(1;\underline 2)^{(1)})\ltimes Fu</Math>, <Math>[W(1;\underline 2),u]=0</Math>, <Math>[\partial^2,u]=\delta u</Math> where <Math>\delta=\{0,1\}</Math>.</Item>
<Item><Math>W(1;\underline 2)\ltimes (F&tlt;h,u&tgt;)</Math>, <Math>[W(1;\underline 2)^{(1)},(F&tlt;h,u&tgt;]=0</Math>, <Math>[h,u]=\delta u</Math>, and
if <Math>\delta=0</Math>, then the action of <Math>x^{(3)}\partial</Math> on
<Math>F&tlt;h,u&tgt;</Math>
is given by one of the following matrices:
<Display>
\left(\begin{array}{cc}
0 & 0\\
0 & 0\end{array}\right),\
\left(\begin{array}{cc}
0 & 1\\
0 & 0\end{array}\right),\
\left(\begin{array}{cc}
1 & 0\\
0 & 1\end{array}\right),\
\left(\begin{array}{cc}
1 & 1\\
0 & 1\end{array}\right),\
\left(\begin{array}{cc}
0 & \xi\\
1 & 1\end{array}\right)\mbox{ where }\xi\in F^*.</Display></Item>
<Item>the algebra is as in (4.), but <Math>\delta=1</Math>.
Note that Theorem 5.1(3/b) of <Cite Key="Strade"/> lists two such algebras
but they turn out to be isomorphic. We take the one with <Math>[x^{(3)}\partial,h]=[x^{(3)}\partial,u]=0</Math>.
</Item>
<Item><Math>W(1;\underline 2)^{(1)}\oplus K</Math> where <Math>K</Math> is a 3-dimensional solvable Lie algebra.</Item>
<Item><Math>W(1;\underline 2)^{(1)}\ltimes \mathcal O(1;\underline 2)/F</Math>.</Item>
<Item>the non-split extension <Math>0\rightarrow \mathcal O(1;\underline 2)/F\rightarrow L\rightarrow W(1;\underline 2)^{(1)}\rightarrow 0</Math>.</Item>
</Enum>
See Theorem 5.1 of <Cite Key="Strade"/>.
</Subsection>
<Subsection Label="appdim6charodd"><Heading>General odd characteristic</Heading>
If the characteristic of the field is odd, then the 6-dimensional non-solvable
Lie algebras are described by Theorems 5.2--5.4 of <Cite Key="Strade"/>.
Over such a field <Math>F</Math>, let us define
the following isomorphism
classes of 6-dimensional non-solvable Lie algebras.
<Enum>
<Item> <Math>\mbox{sl}(2,F)\oplus\mbox{sl}(2,F)
</Math>.</Item>
<Item><Math>\mbox{sl}(2,F_{q^2})</Math> where <Math>F=F_q</Math>;</Item>
<Item><Math>\mbox{sl}(2,F)\oplus K</Math> where <Math>K</Math> is a solvable
Lie algebra with dimension 3;</Item>
<Item><Math>\mbox{sl}(2,F)\ltimes (V(0)\oplus V(1))</Math> where <Math>V(i)</Math> is the <Math>(i+1)</Math>-dimensional irreducible <Math>\mbox{sl}(2,F)</Math>-module;</Item>
<Item><Math>\mbox{sl}(2,F)\ltimes V(2)</Math> where <Math>V(2)</Math> is the <Math>3</Math>-dimensional irreducible <Math>\mbox{sl}(2,F)</Math>-module;
</Item>
<Item><Math>\mbox{sl}(2,F)\ltimes(V(1)\oplus C(L))\cong \mbox{sl}(2,F)\ltimes H</Math> where <Math>H</Math> is the Heisenberg Lie algebra;</Item>
<Item><Math>\mbox{sl}(2,F)\ltimes K</Math> where <Math>K=Fd\oplus K^{(1)}</Math>, <Math>K^{(1)}</Math> is 2-dimensional abelian, isomorphic, as an <Math>\mbox{sl}(2,F)</Math>-module, to <Math>V(1)</Math>, <Math>[\mbox{sl}(2,F),d]=0</Math>, and, for all <Math>v\in K</Math>, <Math>[d,v]=v</Math>;</Item></Enum>
If the characteristic of <Math>F</Math> is at least 7, then these algebras
form a complete and irredundant list of the isomorphism classes of the 6-dimensional non-solvable Lie algebras.
</Subsection>
If the characteristic of the field <Math>F</Math> is 3, then, besides the
classes in Section <Ref Sect="appdim6charodd"/>, we also obtain the following
isomorphism classes.
<Enum>
<Item><Math>\mbox{sl}(2,F)\ltimes V(2,\chi)</Math> where <Math>\chi</Math> is a 3-dimensional character of <Math>\mbox{sl}(2,F)</Math>. Each
such character is described by a field element <Math>\xi</Math> such that
<Math>T^3+T^2-\xi</Math> has a root in <Math>F</Math>; see Proposition 3.5 of
<Cite Key="Strade"/> for more details.
</Item>
<Item><Math>W(1;\underline 1)\ltimes\mathcal O(1;\underline 1)</Math>
where <Math>\mathcal O(1;\underline 1)</Math> is considered as an abelian Lie algebra.
</Item>
<Item><Math>W(1;\underline 1)\ltimes\mathcal O(1;\underline 1)^*</Math> where
<Math>\mathcal O(1;\underline 1)^*</Math> is the dual of <Math>\mathcal O(1;\underline 1)</Math> and it is considered as an abelian Lie algebra.</Item>
<Item>One of the two 6-dimensional central extensions of the non-split
extension <Math>0\rightarrow V(1)\rightarrow L\rightarrow \mbox{sl}(2,F)\rightarrow 0</Math>; see Proposition 4.5 of <Cite Key="Strade"/>.
We note that Proposition 4.5 of <Cite Key="Strade"/> lists
three such central extensions, but one of them is not a Lie algebra.</Item>
<Item>One of the two non-split extensions <Math>0\rightarrow\mbox{rad }
L\rightarrow L\rightarrow L/\mbox{rad } L\rightarrow 0</Math> with a
5-dimensional ideal; see Theorem 5.4 of <Cite Key="Strade"/>.</Item>
</Enum>
We note here that <Cite Key="Strade"/> lists one more non-solvable Lie algebra
over a field of characteristic 3, namely the one in Theorem 5.3(5). However,
this
algebra is isomorphic to the one in Theorem 5.3(4).
</Subsection>
<Subsection Label="appdim6char5"><Heading>Characteristic 5</Heading>
If the characteristic of the field <Math>F</Math> is 5, then, besides the
classes in Section <Ref Sect="appdim6charodd"/>, we also obtain the following
isomorphism classes.
<Enum><Item><Math>W(1;\underline 1)\oplus F</Math>.</Item>
<Item>The non-split central extension <Math>0\rightarrow F\rightarrow L\rightarrow W(1;\underline 1)\rightarrow 0</Math>.</Item></Enum>
</Subsection>
</Section>
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