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<!DOCTYPE html> <html lang="en"> <head> <title>ad -- matrix of the adjoint action</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="Homotopy Lie Algebra of a surjective ring homomorphism" href="index.html">HomotopyLieAlgebra</a> :: <a title="matrix of the adjoint action" href="_ad.html">ad</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="_allgens.html">next</a> | <a href="index.html">previous</a> | <a href="_allgens.html">forward</a> | <a href="index.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>ad -- matrix of the adjoint action</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">M = ad(A,U,e)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">A</span>, <span>an instance of the type <a title="The class of all DGAlgebras" href="../../DGAlgebras/html/___D__G__Algebra.html">DGAlgebra</a></span>, </span></li> <li><span><span class="tt">U</span>, <span>a <a title="the class of all ring elements handled by the engine" href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, linear form in the generators of A</span></li> <li><span><span class="tt">e</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, </span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">M</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, with entries in the ground field</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>The adjoint action of a scalar linear combination of the entries of allgens(A,d-1) U, regarded as an element of Pi^d, acts by bracket multiplication with source Pi^e and target Pi^{d+e}. The output is a matrix whose columns correspond to a generalized row of the output of bracketMatrix. bracketmatrix</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : kk = ZZ/101 o1 = kk o1 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : S = kk[x,y,z] o2 = S o2 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : R = S/ideal(x^2,y^2,z^2-x*y,x*z, y*z) o3 = R o3 : QuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : lastCyclesDegree = 4 o4 = 4</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : KR = koszulComplexDGA(ideal R) o5 = {Ring => S } Underlying algebra => S[T ..T ] 1 5 2 2 2 Differential => {x , y , - x*y + z , x*z, y*z} o5 : DGAlgebra</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : A = acyclicClosure(KR, EndDegree => lastCyclesDegree);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : d = 1 o7 = 1</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : e = 1 o8 = 1</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : U = sum (Gd = allgens(A,d-1)) o9 = x + y + z o9 : S[T ..T ] 1 99</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : ad(A,U,1) o10 = {1, 2} | 2 0 0 | {1, 2} | 0 2 0 | {1, 2} | -1 -1 2 | {1, 2} | 1 0 1 | {1, 2} | 0 1 1 | 5 3 o10 : Matrix (S[T ..T ]) <-- (S[T ..T ]) 1 99 1 99</code></pre> </td> </tr> </table> <div> <p>The columns of this matrix are the functionals that are the sum of the three rows of the bracket multiplication table:</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i11 : matrix{{1,1,1}}*bracketMatrix(A,d,e) o11 = | 2T_1-T_3+T_4 2T_2-T_3+T_5 2T_3+T_4+T_5 | 1 3 o11 : Matrix (S[T ..T ]) <-- (S[T ..T ]) 1 99 1 99</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="Multiplication matrix of the homotopy Lie algebra" href="_bracket__Matrix.html">bracketMatrix</a> -- Multiplication matrix of the homotopy Lie algebra</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">ad</span>:</h2> <ul> <li><kbd>ad(DGAlgebra,RingElement,ZZ)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="matrix of the adjoint action" href="_ad.html">ad</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">HomotopyLieAlgebra.m2:512:0</span>.</p> </div> </div> </div> </body> </html>