GAP (LieAlgDB) - Chapter 3: A description of the Lie algebras that are contained in the package

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3 A description of the Lie algebras that are contained in the package
3.1 Description of the non-solvable Lie algebras
3.2 Dimension 3
3.3 Dimension 4
3.4 Dimension 5
3.4-1 Characteristic 2
3.4-2 Odd characteristic
3.5 Dimension 6
3.5-1 Characteristic 2
3.5-2 General odd characteristic
3.5-3 Characteristic 3
3.5-4 Characteristic 5
3.6 Description of the simple Lie algebras
3.7 Description of the solvable and nilpotent Lie algebras

### 3 A description of the Lie algebras that are contained in the package

#### 3.1 Description of the non-solvable Lie algebras

In this section we list the non-solvable Lie algebras contained in the package. Our notation follows [Str], 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<x_1,\ldots,x_k> denotes the linear subspace generated by x_1,\ldots,x_k, and we also write Fx_1 for F<x_1>

#### 3.2 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:

1. If char F=2 then the algebra is W(1;\underline 2)^{(1)}.

2. If char F>2 then the algebra is \mbox{sl}(2,F).

See Theorem 3.2 of [Str] for details.

#### 3.3 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 [Str]). The classes are as follows:

1. characteristic 2: W(1;\underline 2) and W(1;\underline 2)^{(1)}\oplus F.

2. odd characteristic: \mbox{gl}(2,F).

#### 3.4 Dimension 5

##### 3.4-1 Characteristic 2

Over a finite field F of characteristic 2 there are 5 isomorphism classes of non-solvable Lie algebras with dimension 5:

1. \mbox{Der}(W(1;\underline 2)^{(1)});

2. 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);

3. W(1;\underline 2)^{(1)}\oplus(F\left< h,u\right>), [h,u]=\delta u, where \delta\in\{0,1\} (two algebras).

See Theorem 4.2 of [Str] for details.

##### 3.4-2 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 [Str]). The classes are as follows.

1. \mbox{sl}(2,F)\oplus F<x,y>, [x,y]=\delta y where \delta\in\{0,1\}.

2. \mbox{sl}(2,F)\ltimes V(1) where V(1) is the irreducible 2-dimensional \mbox{sl}(2,F)-module.

3. 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.

4. If \mbox{char }F=5 then there is an additional algebra: W(1;\underline 1).

#### 3.5 Dimension 6

##### 3.5-1 Characteristic 2

Over a field F of characteristic 2, the isomorphism classes of non-solvable Lie algebras are as follows.

1. W(1;\underline 2)^{(1)}\oplus W(1;\underline 2)^{(1)}.

2. W(1;\underline 2)^{(1)}\otimes F_{q^2} where F=F_q.

3. \mbox{Der}(W(1;\underline 2)^{(1)})\ltimes Fu, [W(1;\underline 2),u]=0, [\partial^2,u]=\delta u where \delta=\{0,1\}.

4. W(1;\underline 2)\ltimes (F<h,u>), [W(1;\underline 2)^{(1)},(F<h,u>]=0, [h,u]=\delta u, and if \delta=0, then the action of x^{(3)}\partial on F<h,u> 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^*.

5. the algebra is as in (4.), but \delta=1. Note that Theorem 5.1(3/b) of [Str] 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.

6. W(1;\underline 2)^{(1)}\oplus K where K is a 3-dimensional solvable Lie algebra.

7. W(1;\underline 2)^{(1)}\ltimes \mathcal O(1;\underline 2)/F.

8. 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 [Str].

##### 3.5-2 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 [Str]. Over such a field F, let us define the following isomorphism classes of 6-dimensional non-solvable Lie algebras.

1. \mbox{sl}(2,F)\oplus\mbox{sl}(2,F) .

2. \mbox{sl}(2,F_{q^2}) where F=F_q;

3. \mbox{sl}(2,F)\oplus K where K is a solvable Lie algebra with dimension 3;

4. \mbox{sl}(2,F)\ltimes (V(0)\oplus V(1)) where V(i) is the (i+1)-dimensional irreducible \mbox{sl}(2,F)-module;

5. \mbox{sl}(2,F)\ltimes V(2) where V(2) is the 3-dimensional irreducible \mbox{sl}(2,F)-module;

6. \mbox{sl}(2,F)\ltimes(V(1)\oplus C(L))\cong \mbox{sl}(2,F)\ltimes H where H is the Heisenberg Lie algebra;

7. \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.

##### 3.5-3 Characteristic 3

If the characteristic of the field F is 3, then, besides the classes in Section 3.5-2, we also obtain the following isomorphism classes.

1. \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 [Str] for more details.

2. W(1;\underline 1)\ltimes\mathcal O(1;\underline 1) where \mathcal O(1;\underline 1) is considered as an abelian Lie algebra.

3. 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.

4. 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 [Str]. We note that Proposition 4.5 of [Str] lists three such central extensions, but one of them is not a Lie algebra.

5. 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 [Str].

We note here that [Str] 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).

##### 3.5-4 Characteristic 5

If the characteristic of the field F is 5, then, besides the classes in Section 3.5-2, we also obtain the following isomorphism classes.

1. W(1;\underline 1)\oplus F.

2. The non-split central extension 0\rightarrow F\rightarrow L\rightarrow W(1;\underline 1)\rightarrow 0.

#### 3.6 Description of the simple Lie algebras

If F is a finite field, then, up to isomorphism, there is precisely one simple Lie algebra with dimension 3, and another one with dimension 6; these can be accessed by calling NonSolvableLieAlgebra(F,[3,1]) and NonSolvableLieAlgebra(F,[6,2]) (see NonSolvableLieAlgebra for the details). Over a field of characteristic 5, there is an additional simple Lie algebra with dimension 5, namely NonSolvableLieAlgebra(F,[5,3]). These are the only isomorphism types of simple Lie algebras over finite fields up to dimension 6.

In addition to the algebras above the package contains the simple Lie algebras of dimension between 7 and 9 over GF(2). These Lie algebras were determined by [VL06] and can be described as follows.

There are two isomorphism classes of 7-dimensional Lie algebras over GF(2). In a basis b1,...,b7 the non-trivial products in the first algebra are

    [b1,b2]=b3, [b1,b3]=b4, [b1,b4]=b5, [b1,b5]=b6
    [b1,b6]=b7, [b1,b7]=b1, [b2,b7]=b2, [b3,b6]=b2,
    [b4,b5]=b2, [b4,b6]=b3, [b4,b7]=b4, [b6,b7]=b6;

and those in the second are

    [b1,b2]=b3, [b1,b3]=b1+b4, [b1,b4]=b5, [b1,b5]=b6,
    [b1,b6]=b7, [b2,b3]=b2, [b2,b5]=b2+b4, [b2,b6]=b5,
    [b2,b7]=b1+b4, [b3,b4]=b2+b4, [b3,b5]=b3, [b3,b6]=b1+b4+b6,
    [b3,b7]=b5, [b4,b7]=b6, [b5,b6]=b6, [b5,b7]=b7.

Over GF(2) there are two isomorphism types of simple Lie algebras with dimension 8. In the basis b1,...,b8 the non-trivial products for the first one are

    [b1,b3]=b5, [b1,b4]=b6, [b1,b7]=b2, [b1,b8]=b1, [b2,b3]=b7, [b2,b4]=b5+b8,
    [b2,b5]=b2, [b2,b6]=b1, [b2,b8]=b2, [b3,b6]=b4, [b3,b8]=b3, [b4,b5]=b4,
    [b4,b7]=b3, [b4,b8]=b4, [b5,b6]=b6, [b5,b7]=b7, [b6,b7]=b8;

and for the second one they are

    [b1,b2]=b3, [b1,b3]=b2+b5, [b1,b4]=b6, [b1,b5]=b2, [b1,b6]=b1+b4+b8,
    [b1,b8]=b4, [b2,b3]=b4, [b2,b4]=b1, [b2,b5]=b6, [b2,b6]=b2+b7,
    [b2,b7]=b2+b5, [b3,b4]=b2+b7, [b3,b5]=b1+b4+b8, [b3,b6]=b1, [b3,b7]=b2+b3,
    [b3,b8]=b1, [b4,b5]=b3, [b4,b6]=b2+b4, [b4,b7]=b1+b4+b8, [b4,b8]=b3,
    [b5,b6]=b1+b2+b5, [b5,b7]=b3, [b5,b8]=b2+b7, [b6,b7]=b4+b6, [b6,b8]=b2+b5,
    [b7,b8]=b6.

The non-trivial products for the unique simple Lie algebra with dimension 9 over GF(2) are as follows:

    [b1,b2]=b3, [b1,b3]=b5, [b1,b5]=b6, [b1,b6]=b7, [b1,b7]=b6+b9,
    [b1,b9]=b2, [b2,b3]=b4, [b2,b4]=b6, [b2,b6]=b8, [b2,b8]=b6+b9,
    [b2,b9]=b1, [b3,b4]=b7, [b3,b5]=b8, [b3,b7]=b1+b8, [b3,b8]=b2+b7,
    [b4,b5]=b6+b9, [b4,b6]=b2+b7, [b4,b7]=b3+b6+b9, [b4,b9]=b5,
    [b5,b6]=b1+b8, [b5,b8]=b3+b6+b9, [b5,b9]=b4, [b6,b7]=b1+b4+b8,
    [b6,b8]=b2+b5+b7, [b7,b8]=b3+b9, [b7,b9]=b8, [b8,b9]=b7.

#### 3.7 Description of the solvable and nilpotent Lie algebras

In this section we list the multiplication tables of the nilpotent and solvable Lie algebras contained in the package. Some parametric classes contain isomorphic Lie algebras, for different values of the parameters. For exact descriptions of these isomorphisms we refer to [dG05], [dG07] and [CdGS11]. In dimension 2 there are just two classes of solvable Lie algebras:

- L_2^1: The Abelian Lie algebra.

- L_2^2: [x_2,x_1]=x_1.

We have the following solvable Lie algebras of dimension 3:

- L_3^1 The Abelian Lie algebra.

- L_3^2 [x_3,x_1]=x_1, [x_3,x_2]=x_2.

- L_3^3(a) [x_3,x_1]=x_2, [x_3,x_2]=ax_1+x_2.

- L_3^4(a) [x_3,x_1]=x_2, [x_3,x_2]=ax_1.

And the following solvable Lie algebras of dimension 4:

- L_4^1 The Abelian Lie algebra.

- L_4^2 [x_4,x_1]=x_1, [x_4,x_2]=x_2, [x_4,x_3]=x_3.

- L_4^3(a) [x_4,x_1]=x_1, [x_4,x_2]=x_3, [x_4,x_3]=-ax_2 +(a+1)x_3.

- L_4^4 [x_4,x_2]=x_3, [x_4,x_3]= x_3.

- L_4^5 [x_4,x_2]=x_3.

- L_4^6(a,b) [x_4,x_1] = x_2, [x_4,x_2]=x_3, [x_4,x_3] = ax_1+bx_2+x_3.

- L_4^7(a,b) [x_4,x_1] = x_2, [x_4,x_2]=x_3, [x_4,x_3] = ax_1+bx_2.

- L_4^8 [x_1,x_2]=x_2, [x_3,x_4]=x_4.

- L_4^9(a) [x_4,x_1] = x_1+ax_2, [x_4,x_2]=x_1, [x_3,x_1]=x_1, [x_3,x_2]=x_2.

- L_4^10(a) [x_4,x_1] = x_2, [x_4,x_2]=ax_1, [x_3,x_1]=x_1, [x_3,x_2]=x_2 Condition on F: the characteristic of F is 2.

- L_4^11(a,b) [x_4,x_1] = x_1, [x_4,x_2] = bx_2, [x_4,x_3]=(1+b)x_3, [x_3,x_1]=x_2, [x_3,x_2]=ax_1. Condition on F: the characteristic of F is 2.

- L_4^12 [x_4,x_1] = x_1, [x_4,x_2]=2x_2, [x_4,x_3] = x_3, [x_3,x_1]=x_2.

- L_4^13(a) [x_4,x_1] = x_1+ax_3, [x_4,x_2]=x_2, [x_4,x_3] = x_1, [x_3,x_1]=x_2.

- L_4^14(a) [x_4,x_1] = ax_3, [x_4,x_3]=x_1, [x_3,x_1]=x_2.

Nilpotent of dimension 5:

- N_5,1 Abelian.

- N_5,2 [x_1,x_2]=x_3.

- N_5,3 [x_1,x_2]=x_3, [x_1,x_3]=x_4.

- N_5,4 [x_1,x_2]=x_5, [x_3,x_4]=x_5.

- N_5,5 [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_2,x_4]=x_5.

- N_5,6 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_5.

- N_5,7 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5.

- N_5,8 [x_1,x_2]=x_4, [x_1,x_3]=x_5.

- N_5,9 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_2,x_3]=x_5.

We get nine 6-dimensional nilpotent Lie algebras denoted N_6,k for k=1,...,9 that are the direct sum of N_5,k and a 1-dimensional abelian ideal. Subsequently we get the following Lie algebras.

- N_6,10 [x_1,x_2]=x_3, [x_1,x_3]=x_6, [x_4,x_5]=x_6.

- N_6,11 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_6, [x_2,x_3]=x_6, [x_2,x_5]=x_6.

- N_6,12 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_6, [x_2,x_5]=x_6.

- N_6,13 [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_5]=x_6, [x_2,x_4]=x_5, [x_3,x_4]=x_6.

- N_6,14 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_5, [x_2,x_5]=x_6,[x_3,x_4]=-x_6.

- N_6,15 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_1,x_5]=x_6, [x_2,x_3]=x_5, [x_2,x_4]=x_6.

- N_6,16 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_5]=x_6, [x_3,x_4]=-x_6.

- N_6,17 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_1,x_5]=x_6, [x_2,x_3]= x_6.

- N_6,18 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_1,x_5]=x_6.

- N_6,19(a) [x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_1,x_5]=x_6, [x_2,x_4]=x_6, [x_3,x_5]=a x_6, for a≠0.

- N_6,20 [x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_1,x_5]=x_6, [x_2,x_4]=x_6.

- N_6,21(a) [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_6, [x_2,x_3]=x_5, [x_2,x_5]= a x_6, for a≠0.

- N_6,22(a) [x_1,x_2]=x_5, [x_1,x_3]=x_6, [x_2,x_4]= a x_6, [x_3,x_4]=x_5.

- N_6,23 [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=x_6, [x_2,x_4]= x_5.

- N_6,24(a) [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=a x_6, [x_2,x_3]=x_6, [x_2,x_4]= x_5.

- N_6,25 [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=x_6.

- N_6,26 [x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_3]=x_6.

- N_6,27 [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_2,x_4]= x_6.

- N_6,28 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_6.

- N_6,29 [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_5]=x_6, [x_2,x_4]=x_5+x_6, [x_3,x_4]=x_6, only over fields of characteristic 2.

- N_6,30 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_1,x_5]=x_6, [x_2,x_3]=x_5+x_6, [x_2,x_4]=x_6, only over fields of characteristic 2.

- N_6,31(a) [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]= x_5, [x_2,x_3]=x_5+a x_6, [x_2,x_5]=x_6, [x_3,x_4]=x_6, for a≠0 and only over fields of characteristic 2.

- N_6,32(a) [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]= x_5, [x_2,x_3]=a x_6, [x_2,x_5]=x_6, [x_3,x_4]=x_6, for a≠0 and only over fields of characteristic 2.

- N_6,33 [x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_5]=x_6, [x_3,x_4]=x_6, only over fields of characteristic 2.

- N_6,34 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_5]=x_6, [x_2,x_3]=x_5, [x_2,x_4]=x_6, only over fields of characteristic 2.

- N_6,35(a) [x_1,x_2]=x_5, [x_1,x_3]=x_6, [x_2,x_4]= a x_6, [x_3,x_4]=x_5+x_6, only over fields of characteristic 2.

- N_6,36(a) [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]= a x_6, [x_2,x_3]=x_6, [x_2,x_4]=x_5+x_6, only over fields of characteristic 2.

In [CdGS11], the Lie algebras N_5,k are denoted by L_5,k for all k=1,...,9. Similarly, the Lie algebras N_6,k or N_6,k(a), where k=1,...,36, are denoted by L_6,k or L_6,k(a) if k=1,...,28 and by L_6,k-28^(2) or L_6,k-28^(2)(a) if k=29,...,36.

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