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<Section Label="listdescr">
<Heading>Description of the solvable and nilpotent Lie algebras</Heading>

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 <Cite Key="wdg05"/>,
<Cite Key="wdg07"/> and <Cite Key="cdgs10"/>.

In dimension 2 there are just two classes of solvable Lie algebras:
<List>
<Item> <M>L_2^1</M>: The Abelian Lie algebra. </Item>
<Item> <M>L_2^2</M>: <M>[x_2,x_1]=x_1</M>. </Item>
</List>

We have the following solvable Lie algebras of dimension 3:
<List>
<Item> <M>L_3^1</M> The Abelian Lie algebra. </Item>
<Item> <M>L_3^2</M> <M>[x_3,x_1]=x_1, [x_3,x_2]=x_2</M>. </Item>
<Item> <M>L_3^3(a)</M> <M>[x_3,x_1]=x_2, [x_3,x_2]=ax_1+x_2</M>. </Item>
<Item> <M>L_3^4(a)</M> <M>[x_3,x_1]=x_2, [x_3,x_2]=ax_1. </M> </Item>
</List>

And the following solvable Lie algebras of dimension 4:

<List>
<Item> <M>L_4^1</M> The Abelian Lie algebra. </Item>
<Item> <M>L_4^2</M> <M> [x_4,x_1]=x_1, [x_4,x_2]=x_2, [x_4,x_3]=x_3.</M> </Item>
<Item> <M>L_4^3(a)</M> <M>[x_4,x_1]=x_1, [x_4,x_2]=x_3, [x_4,x_3]=-ax_2 +(a+1)x_3</M>.</Item>
<Item> <M>L_4^4</M> <M>[x_4,x_2]=x_3, [x_4,x_3]= x_3</M>.</Item>
<Item> <M>L_4^5</M> <M>[x_4,x_2]=x_3</M>.</Item>
<Item> <M>L_4^6(a,b)</M> <M>[x_4,x_1] = x_2, [x_4,x_2]=x_3, [x_4,x_3] =
ax_1+bx_2+x_3</M>.</Item>
<Item> <M>L_4^7(a,b)</M> <M>[x_4,x_1] = x_2, [x_4,x_2]=x_3, [x_4,x_3] =
ax_1+bx_2.</M> </Item>
<Item> <M>L_4^8</M> <M>[x_1,x_2]=x_2, [x_3,x_4]=x_4</M>.</Item>
<Item> <M>L_4^9(a)</M> <M>[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</M>.</Item>
<Item> <M>L_4^{10}(a)</M> <M>[x_4,x_1] = x_2, [x_4,x_2]=ax_1, [x_3,x_1]=x_1,
[x_3,x_2]=x_2</M> Condition on F: the characteristic of F is 2.</Item>
<Item> <M>L_4^{11}(a,b)</M> <M>[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</M>.
Condition on F: the characteristic of F is 2. </Item>
<Item> <M>L_4^{12}</M> <M> [x_4,x_1] = x_1, [x_4,x_2]=2x_2, [x_4,x_3] = x_3,
[x_3,x_1]=x_2</M>.</Item>
<Item> <M>L_4^{13}(a)</M> <M>[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</M>.</Item>
<Item> <M>L_4^{14}(a)</M> <M>[x_4,x_1] = ax_3, [x_4,x_3]=x_1, [x_3,x_1]=x_2</M>.</Item>
</List>

Nilpotent of dimension 5:

<List>
<Item> <M>N_{5,1}</M> Abelian. </Item>
<Item> <M>N_{5,2}</M> <M>[x_1,x_2]=x_3</M>.</Item>
<Item> <M>N_{5,3}</M> <M>[x_1,x_2]=x_3, [x_1,x_3]=x_4</M>.</Item>
<Item> <M>N_{5,4}</M> <M>[x_1,x_2]=x_5, [x_3,x_4]=x_5</M>.</Item>
<Item> <M>N_{5,5}</M> <M>[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_2,x_4]=x_5</M>.</Item>
<Item> <M>N_{5,6}</M> <M>[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_5</M>.</Item>
<Item> <M>N_{5,7}</M> <M>[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5</M>.</Item>
<Item> <M>N_{5,8}</M> <M>[x_1,x_2]=x_4, [x_1,x_3]=x_5</M>.</Item>
<Item> <M>N_{5,9}</M> <M>[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_2,x_3]=x_5</M>.</Item>
</List>

We get nine 6-dimensional nilpotent Lie algebras denoted
<M>N_{6,k}</M> for <M>k=1,...,9</M> that are the
direct sum of <M>N_{5,k}</M> and a 1-dimensional abelian ideal. Subsequently
we get the following Lie algebras.

<List>
<Item> <M>N_{6,10}</M> <M>[x_1,x_2]=x_3, [x_1,x_3]=x_6, [x_4,x_5]=x_6.</M> </Item>
<Item> <M>N_{6,11}</M> <M>[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</M>.</Item>
<Item> <M>N_{6,12}</M> <M>[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_6, [x_2,x_5]=x_6</M>.</Item>
<Item> <M>N_{6,13}</M> <M>[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</M>.</Item>
<Item> <M>N_{6,14}</M> <M>[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</M>.</Item>
<Item> <M>N_{6,15}</M> <M>[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</M>.</Item>
<Item> <M>N_{6,16}</M> <M>[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</M>.</Item>
<Item> <M>N_{6,17}</M>  <M>[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</M>.</Item>
<Item> <M>N_{6,18}</M>  <M>[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5,
[x_1,x_5]=x_6</M>.</Item>
<Item> <M>N_{6,19}(a)</M> <M>[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</M>, for <M>a≠0</M>.</Item>
<Item> <M>N_{6,20}</M> <M>[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_1,x_5]=x_6,
[x_2,x_4]=x_6</M>.</Item>
<Item> <M>N_{6,21}(a)</M> <M>[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</M>, for <M>a≠0</M>.</Item>
<Item> <M>N_{6,22}(a)</M> <M>[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</M>.</Item>
<Item> <M>N_{6,23}</M> <M>[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=x_6,
[x_2,x_4]= x_5</M>.</Item>
<Item> <M>N_{6,24}(a)</M> <M>[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</M>.</Item>
<Item> <M>N_{6,25}</M> <M>[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=x_6</M>.</Item>
<Item> <M>N_{6,26}</M> <M>[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_3]=x_6</M>.</Item>
<Item> <M>N_{6,27}</M> <M>[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_2,x_4]= x_6</M>.</Item>
<Item> <M>N_{6,28}</M> <M>[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_6</M>.</Item>
<Item> <M>N_{6,29}</M> <M>[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</M>, only over fields of characteristic <M>2</M>.</Item>
<Item> <M>N_{6,30}</M> <M>[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</M>, only over fields of characteristic <M>2</M>.</Item>
<Item> <M>N_{6,31}(a)</M> <M>[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</M>, for <M>a≠0</M> and only over fields of characteristic <M>2</M>.</Item>
<Item> <M>N_{6,32}(a)</M> <M>[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</M>, for <M>a≠0</M> and only over fields of characteristic <M>2</M>.</Item>
<Item> <M>N_{6,33}</M> <M>[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_5]=x_6, [x_3,x_4]=x_6</M>, only over fields of characteristic <M>2</M>.</Item>
<Item> <M>N_{6,34}</M> <M>[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</M>, only over fields of characteristic <M>2</M>.</Item>
<Item> <M>N_{6,35}(a)</M> <M>[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</M>, only over fields of characteristic <M>2</M>.</Item>
<Item> <M>N_{6,36}(a)</M> <M>[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</M>, only over fields of characteristic <M>2</M>.</Item>
</List>
In <Cite Key="cdgs10"/>, the Lie algebras <M>N_{5,k}</M> are denoted by <M>L_{5,k}</M> for all <M>k=1,...,9</M>. Similarly,
the Lie algebras <M>N_{6,k}</M> or <M>N_{6,k}(a)</M>, where <M>k=1,...,36</M>, are denoted by <M>L_{6,k}</M> or <M>L_{6,k}(a)</M> if <M>k=1,...,28</M> and by <M>L_{6,k-28}^{(2)}</M> or <M>L_{6,k-28}^{(2)}(a)</M> if <M>k=29,...,36</M>.
</Section>

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