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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_rings.html" title="">rings</a> > <a href="___Weyl_spalgebras.html" title="">Weyl algebras</a></div>
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<div><h1>Weyl algebras</h1>
<div>A Weyl algebra is the non-commutative algebra of algebraic differential operators on a polynomial ring. To each variable <tt>x</tt> corresponds the operator <tt>dx</tt> that differentiates with respect to that variable. The evident commutation relation takes the form <tt>dx*x == x*dx + 1</tt>.<p/>
We can give any names we like to the variables in a Weyl algebra, provided we specify the correspondence between the variables and the derivatives, with the <a href="___Weyl__Algebra.html" title="name for an optional argument">WeylAlgebra</a> option, as follows.<p/>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,dx,dy,t,WeylAlgebra => {x=>dx, y=>dy}]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : dx*dy*x*y
o2 = x*y*dx*dy + x*dx + y*dy + 1
o2 : R</pre>
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<tr><td><pre>i3 : dx*x^5
5 4
o3 = x dx + 5x
o3 : R</pre>
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All modules over Weyl algebras are, in Macaulay2, right modules. This means that multiplication of matrices is from the opposite side:<table class="examples"><tr><td><pre>i4 : dx*x
o4 = x*dx + 1
o4 : R</pre>
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<tr><td><pre>i5 : matrix{{dx}} * matrix{{x}}
o5 = | xdx |
1 1
o5 : Matrix R <--- R</pre>
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All Gröbner basis and related computations work over this ring. For an extensive collection of D-module routines (A D-module is a module over a Weyl algebra), see <a href="../../Dmodules/html/index.html" title="algorithms for D-modules">Dmodules</a>.</div>
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