<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- --> <!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html --> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head><title>Dtransposition -- standard transposition for Weyl algebra</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="___Duality.html">next</a> | <a href="___Dtrace_lp__Z__Z_rp.html">previous</a> | <a href="___Duality.html">forward</a> | <a href="___Dtrace_lp__Z__Z_rp.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div> </td> </tr> </table> <hr/> <div><h1>Dtransposition -- standard transposition for Weyl algebra</h1> <div class="single"><h2>Synopsis</h2> <ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>Dtransposition A</tt></div> </dd></dl> </div> </li> <li><div class="single">Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, or <span>a <a href="../../Macaulay2Doc/html/___Chain__Complex.html">chain complex</a></span></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, or <span>a <a href="../../Macaulay2Doc/html/___Chain__Complex.html">chain complex</a></span> of the same type as <tt>A</tt> -- the standard transpose of <tt>A</tt></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div>The standard transposition is the involution of the Weyl algebra which sends <em>x<sup>a</sup>d<sup>b</sup></em> to <em>(-d)<sup>b</sup>x<sup>a</sup></em>. It provides the equivalence in the Weyl algebra between left and right D-modules.<table class="examples"><tr><td><pre>i1 : W = QQ[x,y,Dx,Dy, WeylAlgebra => {x=>Dx,y=>Dy}] o1 = W o1 : PolynomialRing</pre> </td></tr> <tr><td><pre>i2 : L = x^2*Dy + y*Dy^2 + 3*Dx^5*Dy 5 2 2 o2 = 3Dx Dy + x Dy + y*Dy o2 : W</pre> </td></tr> <tr><td><pre>i3 : Dtransposition L 5 2 2 o3 = 3Dx Dy - x Dy + y*Dy + 2Dy o3 : W</pre> </td></tr> </table> </div> </div> <div class="single"><h2>Caveat</h2> <div>The standard transposition of a left ideal should be a right ideal, however M2 currently doesn't support right modules. Thus the output is left ideal generated by the transposition of the previous generators.</div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="../../Macaulay2Doc/html/___Weyl__Algebra.html" title="name for an optional argument">WeylAlgebra</a> -- name for an optional argument</span></li> </ul> </div> <div class="waystouse"><h2>Ways to use <tt>Dtransposition</tt> :</h2> <ul><li>Dtransposition(ChainComplex)</li> <li>Dtransposition(Ideal)</li> <li>Dtransposition(Matrix)</li> <li>Dtransposition(RingElement)</li> </ul> </div> </div> </body> </html>