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<!DOCTYPE html> <html lang="en"> <head> <title>LieElement -- the class of all Lie algebra elements</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="a package for doing computations in graded Lie algebras" href="index.html">GradedLieAlgebras</a> :: <a title="the class of all Lie algebra elements" href="___Lie__Element.html">LieElement</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="___Lie__Element_sp_pl_sp__Lie__Element.html">next</a> | <a href="___Lie__Derivation_sp__Lie__Element.html">previous</a> | <a href="___Lie__Element_sp_pl_sp__Lie__Element.html">forward</a> | <a href="___Lie__Derivation_sp__Lie__Element.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>LieElement -- the class of all Lie algebra elements</h1> <div> <h2>Description</h2> <div> <p>This type represents elements in Lie algebras. Each object of type <span class="tt">LieAlgebra</span> is itself a type <span class="tt">L</span>, and elements in <span class="tt">L</span> belong also to the type <span class="tt">LieElement</span>, which is the parent of <span class="tt">L</span>. Internally an element of type <span class="tt">LieElement</span> is of type <span class="tt">BasicList</span> consisting of two basic lists. The first is a list of coefficients and the second is a list of basic lists of numbered generators that correspond to iterated Lie products of generators.</p> <p></p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : L = lieAlgebra{a,b,c} o1 = L o1 : LieAlgebra</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : showStructure L o2 = Thing : BasicList : LieElement : L o2 : Descent</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : x = a a b - 3 c b a o3 = - (a b a) - 3 (c b a) o3 : L</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : x#0 o4 = BasicList{-1, -3} o4 : BasicList</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : x#1 o5 = BasicList{BasicList{0, 1, 0}, BasicList{2, 1, 0}} o5 : BasicList</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="make a free Lie algebra" href="_lie__Algebra.html">lieAlgebra</a> -- make a free Lie algebra</span></li> <li><span><a title="multiplication of Lie elements" href="___Lie__Element_sp__Lie__Element.html">LieElement LieElement</a> -- multiplication of Lie elements</span></li> <li><span><a href="___First_sp__Lie_spalgebra_sptutorial.html">First Lie algebra tutorial</a></span></li> </ul> </div> <div> <div class="waystouse"> <h2>Functions and methods returning an object of class LieElement:</h2> <ul> <li><span><a title="get the zero element" href="___Z__Z_sp_us_sp__Lie__Algebra.html">ZZ _ LieAlgebra</a> -- get the zero element</span></li> </ul> <h2>Methods that use an object of class LieElement:</h2> <ul> <li><span><a title="unary negation" href="_-_sp__Lie__Element.html">- LieElement</a> -- unary negation</span></li> <li><span><a title="get the coefficients and monomials of a Lie element" href="_coefficients_lp__Lie__Element_rp.html">coefficients(LieElement)</a> -- get the coefficients and monomials of a Lie element</span></li> <li><span><kbd>firstDegree(LieElement)</kbd> -- see <span><a title="get the degree of an element" href="_first__Degree.html">firstDegree</a> -- get the degree of an element</span></span></li> <li><span><kbd>indexForm(LieElement)</kbd> -- see <span><a title="get a Lie element in the polynomial ring representation" href="_index__Form.html">indexForm</a> -- get a Lie element in the polynomial ring representation</span></span></li> <li><span><kbd>innerDerivation(LieElement)</kbd> -- see <span><a title="make the derivation defined by right Lie multiplication by a Lie element" href="_inner__Derivation.html">innerDerivation</a> -- make the derivation defined by right Lie multiplication by a Lie element</span></span></li> <li><span><a title="whether a Lie element belongs to a Lie subspace" href="_is__Member_lp__Lie__Element_cm__Lie__Sub__Space_rp.html">isMember(LieElement,LieSubSpace)</a> -- whether a Lie element belongs to a Lie subspace</span></li> <li><span><a title="formal application of a Lie map to a Lie element" href="___Lie__Algebra__Map_sp_at_sp__Lie__Element.html">LieAlgebraMap @ LieElement</a> -- formal application of a Lie map to a Lie element</span></li> <li><span><a title="apply a Lie homomorphism" href="___Lie__Algebra__Map_sp__Lie__Element.html">LieAlgebraMap LieElement</a> -- apply a Lie homomorphism</span></li> <li><span><a title="formal application of a derivation to a Lie element" href="___Lie__Derivation_sp_at_sp__Lie__Element.html">LieDerivation @ LieElement</a> -- formal application of a derivation to a Lie element</span></li> <li><span><a title="apply a derivation" href="___Lie__Derivation_sp__Lie__Element.html">LieDerivation LieElement</a> -- apply a derivation</span></li> <li><span><a title="addition of Lie elements" href="___Lie__Element_sp_pl_sp__Lie__Element.html">LieElement + LieElement</a> -- addition of Lie elements</span></li> <li><span><a title="formal addition of Lie elements" href="___Lie__Element_sp_pl_pl_sp__Lie__Element.html">LieElement ++ LieElement</a> -- formal addition of Lie elements</span></li> <li><span><a title="subtraction of Lie elements" href="___Lie__Element_sp-_sp__Lie__Element.html">LieElement - LieElement</a> -- subtraction of Lie elements</span></li> <li><span><a title="formal subtraction of Lie elements" href="___Lie__Element_sp_sl_sp__Lie__Element.html">LieElement / LieElement</a> -- formal subtraction of Lie elements</span></li> <li><span><a title="formal multiplication of Lie elements" href="___Lie__Element_sp_at_sp__Lie__Element.html">LieElement @ LieElement</a> -- formal multiplication of Lie elements</span></li> <li><span><a title="multiplication of Lie elements" href="___Lie__Element_sp__Lie__Element.html">LieElement LieElement</a> -- multiplication of Lie elements</span></li> <li><span><a title="get the monomials of a Lie element" href="_monomials_lp__Lie__Element_rp.html">monomials(LieElement)</a> -- get the monomials of a Lie element</span></li> <li><span><kbd>normalForm(LieElement)</kbd> -- see <span><a title="compute the normal form of a LieElement" href="_normal__Form.html">normalForm</a> -- compute the normal form of a LieElement</span></span></li> <li><span><a title="formal multiplication of a number and a Lie element" href="___Number_sp_at_sp__Lie__Element.html">Number @ LieElement</a> -- formal multiplication of a number and a Lie element</span></li> <li><span><a title="multiplication of a number and a Lie element" href="___Number_sp__Lie__Element.html">Number LieElement</a> -- multiplication of a number and a Lie element</span></li> <li><span><a title="formal multiplication of a ring element and a Lie element" href="___Ring__Element_sp_at_sp__Lie__Element.html">RingElement @ LieElement</a> -- formal multiplication of a ring element and a Lie element</span></li> <li><span><a title="multiplication of a field element and a Lie element" href="___Ring__Element_sp__Lie__Element.html">RingElement LieElement</a> -- multiplication of a field element and a Lie element</span></li> <li><span><a title="get the sign of a homogeneous element " href="_sign_lp__Lie__Element_rp.html">sign(LieElement)</a> -- get the sign of a homogeneous element </span></li> <li><span><kbd>weight(LieElement)</kbd> -- see <span><a title="get the weight of a homogeneous element" href="_weight.html">weight</a> -- get the weight of a homogeneous element</span></span></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="the class of all Lie algebra elements" href="___Lie__Element.html">LieElement</a> is <span>a <a title="the class of all mutable types" href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a title="the class of all basic lists" href="../../Macaulay2Doc/html/___Basic__List.html">BasicList</a> < <a title="the class of all things" href="../../Macaulay2Doc/html/___Thing.html">Thing</a>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">GradedLieAlgebras/doc.m2:141:0</span>.</p> </div> </div> </div> </body> </html>