<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (HAP) - Chapter 10: Lie commutators and nonabelian Lie tensors</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> </head> <body><a href="../www/index.html"><small>HAP home</small></a> <div class="chlinkprevnexttop"> <a href="chap0.html">Top of Book</a> <a href="chap9.html">Previous Chapter</a> <a href="chap11.html">Next Chapter</a> </div> <p><a id="X7A3DC9327EE1BE6C" name="X7A3DC9327EE1BE6C"></a></p> <div class="ChapSects"><a href="chap10.html#X7A3DC9327EE1BE6C">10. <span class="Heading"> Lie commutators and nonabelian Lie tensors</span></a> </div> <h3>10. <span class="Heading"> Lie commutators and nonabelian Lie tensors</span></h3> <div class="pcenter"><table cellspacing="10" class="GAPDocTable"> <tr> <td class="tdleft">Functions on this page are joint work with Hamid Mohammadzadeh, and implemented by him.</td> </tr> <tr> <td class="tdleft"><code class="code"> LieCoveringHomomorphism(L)</code></p> <p>Inputs a finite dimensional Lie algebra L over a field, and returns a surjective Lie homomorphism phi : C-> L where:</p> <ul> <li><p>the kernel of phi lies in both the centre of C and the derived subalgebra of C,</p> </li> <li><p>the kernel of phi is a vector space of rank equal to the rank of the second Chevalley-Eilenberg homology of L.</p> </li> </ul> </td> </tr> <tr> <td class="tdleft"><code class="code"> LeibnizQuasiCoveringHomomorphism(L)</code></p> <p>Inputs a finite dimensional Lie algebra L over a field, and returns a surjective homomorphism phi : C-> L of Leibniz algebras where:</p> <ul> <li><p>the kernel of phi lies in both the centre of C and the derived subalgebra of C,</p> </li> <li><p>the kernel of phi is a vector space of rank equal to the rank of the kernel J of the homomorphism L otimes L -> L from the tensor square to L. (We note that, in general, J is NOT equal to the second Leibniz homology of L.)</p> </li> </ul> </td> </tr> <tr> <td class="tdleft"><code class="code"> LieEpiCentre(L)</code></p> <p>Inputs a finite dimensional Lie algebra L over a field, and returns an ideal Z^*(L) of the centre of L. The ideal Z^*(L) is trivial if and only if L is isomorphic to a quotient L=E/Z(E) of some Lie algebra E by the centre of E.</td> </tr> <tr> <td class="tdleft"><code class="code"> LieExteriorSquare(L) </code></p> <p>Inputs a finite dimensional Lie algebra L over a field. It returns a record E with the following components.</p> <ul> <li><p>E.homomorphism is a Lie homomorphism µ : (L wedge L) --> L from the nonabelian exterior square (L wedge L) to L. The kernel of µ is the Lie multiplier.</p> </li> <li><p>E.pairing(x,y) is a function which inputs elements x, y in L and returns (x wedge y) in the exterior square (L wedge L) .</p> </li> </ul> </td> </tr> <tr> <td class="tdleft"><code class="code"> LieTensorSquare(L) </code></p> <p>Inputs a finite dimensional Lie algebra L over a field and returns a record T with the following components.</p> <ul> <li><p>T.homomorphism is a Lie homomorphism µ : (L otimes L) --> L from the nonabelian tensor square of L to L.</p> </li> <li><p>T.pairing(x,y) is a function which inputs two elements x, y in L and returns the tensor (x otimes y) in the tensor square (L otimes L) .</p> </li> </ul> </td> </tr> <tr> <td class="tdleft"><code class="code"> LieTensorCentre(L) </code></p> <p>Inputs a finite dimensional Lie algebra L over a field and returns the largest ideal N such that the induced homomorphism of nonabelian tensor squares (L otimes L) --> (L/N otimes L/N) is an isomorphism.</td> </tr> </table><br /><p> </p><br /> </div> <div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</a> <a href="chap9.html">Previous Chapter</a> <a href="chap11.html">Next Chapter</a> </div> <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chapInd.html">Ind</a> </div> <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body> </html>