<h4>3.1 <span class="Heading">Description of the non-solvable Lie algebras</span></h4>
<p>In this section we list the non-solvable Lie algebras contained in the package. Our notation follows <a href="chapBib_mj.html#biBStrade">[Str]</a>, where a more detailed description can also be found. In particular if <span class="SimpleMath">\(L\)</span> is a Lie algebra over <span class="SimpleMath">\(F\)</span> then <span class="SimpleMath">\(C(L)\)</span> denotes the center of <span class="SimpleMath">\(L\)</span>. Further, if <span class="SimpleMath">\(x_1,\ldots,x_k\)</span> are elements of <span class="SimpleMath">\(L\)</span>, then <span class="SimpleMath">\(F<x_1,\ldots,x_k>\)</span> denotes the linear subspace generated by <span class="SimpleMath">\(x_1,\ldots,x_k\)</span>, and we also write <span class="SimpleMath">\(Fx_1\)</span> for <span class="SimpleMath">\(F<x_1>\)</span></p>
<p>There are no non-solvable Lie algebras with dimension 1 or 2. Over an arbitrary finite field <var class="Arg">F</var>, there is just one isomorphism type of non-solvable Lie algebras:</p>
<ol>
<li><p>If <var class="Arg">char F=2</var> then the algebra is <span class="SimpleMath">\(W(1;\underline 2)^{(1)}\)</span>.</p>
</li>
<li><p>If <var class="Arg">char F>2</var> then the algebra is <span class="SimpleMath">\(\mbox{sl}(2,F)\)</span>.</p>
</li>
</ol>
<p>See Theorem 3.2 of <a href="chapBib_mj.html#biBStrade">[Str]</a> for details.</p>
<p>Over a finite field <var class="Arg">F</var> of characteristic 2 there are two isomorphism classes of non-solvable Lie algebras with dimension 4, while over a finite field <var class="Arg">F</var> of odd characteristic the number of isomorphism classes is one (see Theorem 4.1 of <a href="chapBib_mj.html#biBStrade">[Str]</a>). The classes are as follows:</p>
<p>Over a field <span class="SimpleMath">\(F\)</span>of odd characteristic the number of isomorphism types of 5-dimensional non-solvable Lie algebras is <span class="SimpleMath">\(3\)</span> if the characteristic is at least 7, and it is 4 otherwise (see Theorem 4.3 of <a href="chapBib_mj.html#biBStrade">[Str]</a>). The classes are as follows.</p>
<ol>
<li><p><span class="SimpleMath">\(\mbox{sl}(2,F)\oplus F<x,y>\)</span>, <span class="SimpleMath">\([x,y]=\delta y\)</span> where <span class="SimpleMath">\(\delta\in\{0,1\}\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\(\mbox{sl}(2,F)\ltimes V(1)\)</span> where <span class="SimpleMath">\(V(1)\)</span> is the irreducible 2-dimensional <span class="SimpleMath">\(\mbox{sl}(2,F)\)</span>-module.</p>
</li>
<li><p>If <span class="SimpleMath">\(\mbox{char }F=3\)</span> then there is an additional algebra, namely the non-split extension <span class="SimpleMath">\(0\rightarrow V(1)\rightarrow L\rightarrow\mbox{sl}(2,F)\rightarrow 0\)</span>.</p>
</li>
<li><p>If <span class="SimpleMath">\(\mbox{char }F=5\)</span> then there is an additional algebra: <span class="SimpleMath">\(W(1;\underline 1)\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\(W(1;\underline 2)\ltimes (F<h,u>)\)</span>, <span class="SimpleMath">\([W(1;\underline 2)^{(1)},(F<h,u>]=0\)</span>, <span class="SimpleMath">\([h,u]=\delta u\)</span>, and if <span class="SimpleMath">\(\delta=0\)</span>, then the action of <span class="SimpleMath">\(x^{(3)}\partial\)</span> on <span class="SimpleMath">\(F<h,u>\)</span> is given by one of the following matrices:</p>
</li>
<li><p>the algebra is as in (4.), but <span class="SimpleMath">\(\delta=1\)</span>. Note that Theorem 5.1(3/b) of <a href="chapBib_mj.html#biBStrade">[Str]</a> lists two such algebras but they turn out to be isomorphic. We take the one with <span class="SimpleMath">\([x^{(3)}\partial,h]=[x^{(3)}\partial,u]=0\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\(W(1;\underline 2)^{(1)}\oplus K\)</span> where <span class="SimpleMath">\(K\)</span> is a 3-dimensional solvable Lie algebra.</p>
<p>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 <a href="chapBib_mj.html#biBStrade">[Str]</a>. Over such a field <span class="SimpleMath">\(F\)</span>, let us define the following isomorphism classes of 6-dimensional non-solvable Lie algebras.</p>
</li>
<li><p><span class="SimpleMath">\(\mbox{sl}(2,F_{q^2})\)</span> where <span class="SimpleMath">\(F=F_q\)</span>;</p>
</li>
<li><p><span class="SimpleMath">\(\mbox{sl}(2,F)\oplus K\)</span> where <span class="SimpleMath">\(K\)</span> is a solvable Lie algebra with dimension 3;</p>
</li>
<li><p><span class="SimpleMath">\(\mbox{sl}(2,F)\ltimes (V(0)\oplus V(1))\)</span> where <spanclass="SimpleMath">\(V(i)\)</span> is the <span class="SimpleMath">\((i+1)\)</span>-dimensional irreducible <span class="SimpleMath">\(\mbox{sl}(2,F)\)</span>-module;</p>
</li>
<li><p><span class="SimpleMath">\(\mbox{sl}(2,F)\ltimes V(2)\)</span> where <span class="SimpleMath">\(V(2)\)</span> is the <span class="SimpleMath">\(3\)</span>-dimensional irreducible <span class="SimpleMath">\(\mbox{sl}(2,F)\)</span>-module;</p>
</li>
<li><p><span class="SimpleMath">\(\mbox{sl}(2,F)\ltimes(V(1)\oplus C(L))\cong \mbox{sl}(2,F)\ltimes H\)</span> where <span class="SimpleMath">\(H\)</span> is the Heisenberg Lie algebra;</p>
</li>
<li><p><span class="SimpleMath">\(\mbox{sl}(2,F)\ltimes K\)</span> where <span class="SimpleMath">\(K=Fd\oplus K^{(1)}\)</span>, <span class="SimpleMath">\(K^{(1)}\)</span> is 2-dimensional abelian, isomorphic, as an <span class="SimpleMath">\(\mbox{sl}(2,F)\)</span>-module, to <span class="SimpleMath">\(V(1)\)</span>, <span class="SimpleMath">\([\mbox{sl}(2,F),d]=0\)</span>, and, for all <span class="SimpleMath">\(v\in K\)</span>, <span class="SimpleMath">\([d,v]=v\)</span>;</p>
</li>
</ol>
<p>If the characteristic of <span class="SimpleMath">\(F\)</span> 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.</p>
<p>If the characteristic of the field <span class="SimpleMath">\(F\)</span> is 3, then, besides the classes in Section <a href="chap3_mj.html#X7CC42FD178D384FD"><span class="RefLink">3.5-2</span></a>, we also obtain the following isomorphism classes.</p>
<ol>
<li><p><span class="SimpleMath">\(\mbox{sl}(2,F)\ltimes V(2,\chi)\)</span> where <span class="SimpleMath">\(\chi\)</span> is a 3-dimensional character of <span class="SimpleMath">\(\mbox{sl}(2,F)\)</span>. Each such character is described by a field element <span class="SimpleMath">\(\xi\)</span> such that <span class="SimpleMath">\(T^3+T^2-\xi\)</span> has a root in <span class="SimpleMath">\(F\)</span>; see Proposition 3.5 of <a href="chapBib_mj.html#biBStrade">[Str]</a> for more details.</p>
</li>
<li><p><span class="SimpleMath">\(W(1;\underline 1)\ltimes\mathcal O(1;\underline 1)\)</span> where <span class="SimpleMath">\(\mathcal O(1;\underline 1)\)</span> is considered as an abelian Lie algebra.</p>
</li>
<li><p><span class="SimpleMath">\(W(1;\underline 1)\ltimes\mathcal O(1;\underline 1)^*\)</span> where <span class="SimpleMath">\(\mathcal O(1;\underline 1)^*\)</span> is the dual of <span class="SimpleMath">\(\mathcal O(1;\underline 1)\)</span> and it is considered as an abelian Lie algebra.</p>
</li>
<li><p>One of the two 6-dimensional central extensions of the non-split extension <span class="SimpleMath">\(0\rightarrow V(1)\rightarrow L\rightarrow \mbox{sl}(2,F)\rightarrow 0\)</span>; see Proposition 4.5 of <a href="chapBib_mj.html#biBStrade">[Str]</a>. We note that Proposition 4.5 of <a href="chapBib_mj.html#biBStrade">[Str]</a> lists three such central extensions, but one of them is not a Lie algebra.</p>
</li>
<li><p>One of the two non-split extensions <span class="SimpleMath">\(0\rightarrow\mbox{rad } L\rightarrow L\rightarrow L/\mbox{rad } L\rightarrow 0\)</span> with a 5-dimensional ideal; see Theorem 5.4 of <a href="chapBib_mj.html#biBStrade">[Str]</a>.</p>
</li>
</ol>
<p>We note here that <a href="chapBib_mj.html#biBStrade">[Str]</a> 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).</p>
<p>If the characteristic of the field <span class="SimpleMath">\(F\)</span> is 5, then, besides the classes in Section <a href="chap3_mj.html#X7CC42FD178D384FD"><span class="RefLink">3.5-2</span></a>, we also obtain the following isomorphism classes.</p>
<h4>3.6 <span class="Heading">Description of the simple Lie algebras</span></h4>
<p>If <var class="Arg">F</var> 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 <var class="Arg">NonSolvableLieAlgebra(F,[3,1])</var> and <var class="Arg">NonSolvableLieAlgebra(F,[6,2])</var> (see <var class="Arg">NonSolvableLieAlgebra</var> for the details). Over a field of characteristic 5, there is an additional simple Lie algebra with dimension 5, namely <var class="Arg">NonSolvableLieAlgebra(F,[5,3])</var>. These are the only isomorphism types of simple Lie algebras over finite fields up to dimension 6.</p>
<p>In addition to the algebras above the package contains the simple Lie algebras of dimension between 7 and 9 over <var class="Arg">GF(2)</var>. These Lie algebras were determined by <a href="chapBib_mj.html#biBVL">[Vau06]</a> and can be described as follows.</p>
<p>There are two isomorphism classes of 7-dimensional Lie algebras over <var class="Arg">GF(2)</var>. In a basis <span class="SimpleMath">\(b1,\ldots,b7\)</span> the non-trivial products in the first algebra are</p>
<p>Over <var class="Arg">GF(2)</var> there are two isomorphism types of simple Lie algebras with dimension 8. In the basis <span class="SimpleMath">\(b1,\ldots,b8\)</span> the non-trivial products for the first one are</p>
<h4>3.7 <span class="Heading">Description of the solvable and nilpotent Lie algebras</span></h4>
<p>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 <a href="chapBib_mj.html#biBwdg05">[dG05]</a>, <a href="chapBib_mj.html#biBwdg07">[dG07]</a> and <a href="chapBib_mj.html#biBcdgs10">[CdGS11]</a>. In dimension 2 there are just two classes of solvable Lie algebras:</p>
<ul>
<li><p><span class="SimpleMath">\(L_2^1\)</span>: The Abelian Lie algebra.</p>
</li>
<li><p><span class="SimpleMath">\(L_4^{10}(a)\)</span> <span class="SimpleMath">\([x_4,x_1] = x_2, [x_4,x_2]=ax_1, [x_3,x_1]=x_1, [x_3,x_2]=x_2\)</span> Condition on F: the characteristic of F is 2.</p>
</li>
<li><p><span class="SimpleMath">\(L_4^{11}(a,b)\)</span> <span class="SimpleMath">\([x_4,x_1] = x_1, [x_4,x_2] = bx_2, [x_4,x_3]=(1+b)x_3, [x_3,x_1]=x_2, [x_3,x_2]=ax_1\)</span>. Condition on F: the characteristic of F is 2.</p>
</li>
</ul>
<p>We get nine 6-dimensional nilpotent Lie algebras denoted <span class="SimpleMath">\(N_{6,k}\)</span> for <span class="SimpleMath">\(k=1,...,9\)</span> that are the direct sum of <span class="SimpleMath">\(N_{5,k}\)</span> and a 1-dimensional abelian ideal. Subsequently we get the following Lie algebras.</p>
</li>
<li><p><span class="SimpleMath">\(N_{6,29}\)</span> <span class="SimpleMath">\([x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_5]=x_6, [x_2,x_4]=x_5+x_6, [x_3,x_4]=x_6\)</span>, only over fields of characteristic <span class="SimpleMath">\(2\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\(N_{6,30}\)</span> <span class="SimpleMath">\([x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_1,x_5]=x_6, [x_2,x_3]=x_5+x_6, [x_2,x_4]=x_6\)</span>, only over fields of characteristic <span class="SimpleMath">\(2\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\(N_{6,31}(a)\)</span> <span class="SimpleMath">\([x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]= x_5, [x_2,x_3]=x_5+a x_6, [x_2,x_5]=x_6, [x_3,x_4]=x_6\)</span>, for <span class="SimpleMath">\(a≠0\)</span> and only over fields of characteristic <span class="SimpleMath">\(2\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\(N_{6,32}(a)\)</span> <span class="SimpleMath">\([x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]= x_5, [x_2,x_3]=a x_6, [x_2,x_5]=x_6, [x_3,x_4]=x_6\)</span>, for <span class="SimpleMath">\(a≠0\)</span> and only over fields of characteristic <span class="SimpleMath">\(2\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\(N_{6,33}\)</span> <span class="SimpleMath">\([x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_5]=x_6, [x_3,x_4]=x_6\)</span>, only over fields of characteristic <span class="SimpleMath">\(2\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\(N_{6,34}\)</span> <span class="SimpleMath">\([x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_5]=x_6, [x_2,x_3]=x_5, [x_2,x_4]=x_6\)</span>, only over fields of characteristic <span class="SimpleMath">\(2\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\(N_{6,35}(a)\)</span> <span class="SimpleMath">\([x_1,x_2]=x_5, [x_1,x_3]=x_6, [x_2,x_4]= a x_6, [x_3,x_4]=x_5+x_6\)</span>, only over fields of characteristic <span class="SimpleMath">\(2\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\(N_{6,36}(a)\)</span> <span class="SimpleMath">\([x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]= a x_6, [x_2,x_3]=x_6, [x_2,x_4]=x_5+x_6\)</span>, only over fields of characteristic <span class="SimpleMath">\(2\)</span>.</p>
</li>
</ul>
<p>In <a href="chapBib_mj.html#biBcdgs10">[CdGS11]</a>, the Lie algebras <span class="SimpleMath">\(N_{5,k}\)</span> are denoted by <span class="SimpleMath">\(L_{5,k}\)</span> for all <span class="SimpleMath">\(k=1,...,9\)</span>. Similarly, the Lie algebras <span class="SimpleMath">\(N_{6,k}\)</span> or <span class="SimpleMath">\(N_{6,k}(a)\)</span>, where <span class="SimpleMath">\(k=1,...,36\)</span>, are denoted by <span class="SimpleMath">\(L_{6,k}\)</span> or <span class="SimpleMath">\(L_{6,k}(a)\)</span> if <span class="SimpleMath">\(k=1,...,28\)</span> and by <span class="SimpleMath">\(L_{6,k-28}^{(2)}\)</span> or <span class="SimpleMath">\(L_{6,k-28}^{(2)}(a)\)</span> if <span class="SimpleMath">\(k=29,...,36\)</span>.</p>
¤ Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.0.21Bemerkung:
(vorverarbeitet)
¤
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
SSL
Die farbliche Syntaxdarstellung ist noch experimentell.
