Description of the non-solvable Lie algebras

In this section we list the non-solvable Lie algebras contained in the package. Our notation follows , where a more detailed description can also be found. In particular if L is a Lie algebra over F then C(L) denotes the center of L. Further, if x_1,\ldots,x_k are elements of L, then F&tlt;x_1,\ldots,x_k&tgt; denotes the linear subspace generated by x_1,\ldots,x_k, and we also write Fx_1 for F&tlt;x_1&tgt;
Dimension 3

There are no non-solvable Lie algebras with dimension 1 or 2. Over an arbitrary finite field F, there is just one isomorphism type of non-solvable Lie algebras: If char F=2 then the algebra is W(1;\underline 2)^{(1)}. If char F>2 then the algebra is \mbox{sl}(2,F). See Theorem 3.2 of for details.
Dimension 4
Over a finite field F of characteristic 2 there are two isomorphism classes of non-solvable Lie algebras with dimension 4, while over a finite field F of odd characteristic the number of isomorphism classes is one (see Theorem 4.1 of ). The classes are as follows: characteristic 2: W(1;\underline 2) and W(1;\underline 2)^{(1)}\oplus F. odd characteristic: \mbox{gl}(2,F).
Dimension 5
Characteristic 2
Over a finite field F of characteristic 2 there are 5 isomorphism classes of non-solvable Lie algebras with dimension 5: \mbox{Der}(W(1;\underline 2)^{(1)}); W(1;\underline 2)\ltimes Fu where [W(1;\underline 2)^{(1)},u]=0, [x^{(3)}\partial,u]=\delta u and \delta\in\{0,1\} (two algebras); W(1;\underline 2)^{(1)}\oplus(F\left&tlt; h,u\right&tgt;), [h,u]=\delta u, where \delta\in\{0,1\} (two algebras). See Theorem 4.2 of for details.
Odd characteristic
Over a field Fof odd characteristic the number of isomorphism types of 5-dimensional non-solvable Lie algebras is 3 if the characteristic is at least 7, and it is 4 otherwise (see Theorem 4.3 of ). The classes are as follows. \mbox{sl}(2,F)\oplus F&tlt;x,y&tgt;, [x,y]=\delta y where \delta\in\{0,1\}. \mbox{sl}(2,F)\ltimes V(1) where V(1) is the irreducible 2-dimensional \mbox{sl}(2,F)-module. If \mbox{char }F=3 then there is an additional algebra, namely the non-split extension 0\rightarrow V(1)\rightarrow L\rightarrow\mbox{sl}(2,F)\rightarrow 0. If \mbox{char }F=5 then there is an additional algebra: W(1;\underline 1).
Dimension 6
Characteristic 2
Over a field F of characteristic 2, the isomorphism classes of non-solvable Lie algebras are as follows. W(1;\underline 2)^{(1)}\oplus W(1;\underline 2)^{(1)}. W(1;\underline 2)^{(1)}\otimes F_{q^2} where F=F_q. \mbox{Der}(W(1;\underline 2)^{(1)})\ltimes Fu, [W(1;\underline 2),u]=0, [\partial^2,u]=\delta u where \delta=\{0,1\}. W(1;\underline 2)\ltimes (F&tlt;h,u&tgt;), [W(1;\underline 2)^{(1)},(F&tlt;h,u&tgt;]=0, [h,u]=\delta u, and if \delta=0, then the action of x^{(3)}\partial on F&tlt;h,u&tgt; is given by one of the following matrices: \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^*. the algebra is as in (4.), but \delta=1. Note that Theorem 5.1(3/b) of lists two such algebras but they turn out to be isomorphic. We take the one with [x^{(3)}\partial,h]=[x^{(3)}\partial,u]=0. W(1;\underline 2)^{(1)}\oplus K where K is a 3-dimensional solvable Lie algebra. W(1;\underline 2)^{(1)}\ltimes \mathcal O(1;\underline 2)/F. the non-split extension 0\rightarrow \mathcal O(1;\underline 2)/F\rightarrow L\rightarrow W(1;\underline 2)^{(1)}\rightarrow 0. See Theorem 5.1 of .
General odd characteristic
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 . Over such a field F, let us define the following isomorphism classes of 6-dimensional non-solvable Lie algebras. \mbox{sl}(2,F)\oplus\mbox{sl}(2,F) . \mbox{sl}(2,F_{q^2}) where F=F_q; \mbox{sl}(2,F)\oplus K where K is a solvable Lie algebra with dimension 3; \mbox{sl}(2,F)\ltimes (V(0)\oplus V(1)) where V(i) is the (i+1)-dimensional irreducible \mbox{sl}(2,F)-module; \mbox{sl}(2,F)\ltimes V(2) where V(2) is the 3-dimensional irreducible \mbox{sl}(2,F)-module; \mbox{sl}(2,F)\ltimes(V(1)\oplus C(L))\cong \mbox{sl}(2,F)\ltimes H where H is the Heisenberg Lie algebra; \mbox{sl}(2,F)\ltimes K where K=Fd\oplus K^{(1)}, K^{(1)} is 2-dimensional abelian, isomorphic, as an \mbox{sl}(2,F)-module, to V(1), [\mbox{sl}(2,F),d]=0, and, for all v\in K, [d,v]=v;

If the characteristic of F 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.
Characteristic 3

If the characteristic of the field F is 3, then, besides the classes in Section , we also obtain the following isomorphism classes. \mbox{sl}(2,F)\ltimes V(2,\chi) where \chi is a 3-dimensional character of \mbox{sl}(2,F). Each such character is described by a field element \xi such that T^3+T^2-\xi has a root in F; see Proposition 3.5 of for more details. W(1;\underline 1)\ltimes\mathcal O(1;\underline 1) where \mathcal O(1;\underline 1) is considered as an abelian Lie algebra. W(1;\underline 1)\ltimes\mathcal O(1;\underline 1)^* where \mathcal O(1;\underline 1)^* is the dual of \mathcal O(1;\underline 1) and it is considered as an abelian Lie algebra. One of the two 6-dimensional central extensions of the non-split extension 0\rightarrow V(1)\rightarrow L\rightarrow \mbox{sl}(2,F)\rightarrow 0; see Proposition 4.5 of . We note that Proposition 4.5 of lists three such central extensions, but one of them is not a Lie algebra. One of the two non-split extensions 0\rightarrow\mbox{rad } L\rightarrow L\rightarrow L/\mbox{rad } L\rightarrow 0 with a 5-dimensional ideal; see Theorem 5.4 of .

We note here that 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).
Characteristic 5
If the characteristic of the field F is 5, then, besides the classes in Section , we also obtain the following isomorphism classes. W(1;\underline 1)\oplus F. The non-split central extension 0\rightarrow F\rightarrow L\rightarrow W(1;\underline 1)\rightarrow 0.