Editing Weyl algebra (section)

{{Short description|Differential algebra}}
In [[abstract algebra]], the '''Weyl algebras''' are abstracted from the [[ring (mathematics)|ring]] of [[differential operator]]s with [[polynomial]] coefficients. They are named after [[Hermann Weyl]], who introduced them to study the [[Werner Heisenberg|Heisenberg]] [[uncertainty principle]] in [[quantum mechanics]]. 

In the simplest case, these are differential operators. Let <math>F</math> be a [[field (mathematics)|field]], and let <math>F[x]</math> be the [[polynomial ring|ring of polynomials]] in one variable with coefficients in <math>F</math>. Then the corresponding Weyl algebra consists of differential operators of form

: <math> f_m(x) \partial_x^m + f_{m-1}(x) \partial_x^{m-1} + \cdots + f_1(x) \partial_x + f_0(x) </math>

This is the '''first Weyl algebra''' <math>A_1</math>. The '''''n''-th Weyl algebra''' <math>A_n</math> are constructed similarly. 

Alternatively, <math>A_1</math> can be constructed as the [[quotient ring|quotient]] of the [[free algebra]] on two generators, ''q'' and ''p'', by the [[ideal (ring theory)|ideal]] generated by <math>([p,q] - 1)</math>. Similarly, <math>A_n</math> is obtained by quotienting the free algebra on ''2n'' generators by the ideal generated by<math display="block"> ([p_i,q_j] - \delta_{i,j}), \quad \forall i, j = 1, \dots, n</math>where <math> \delta_{i,j}</math> is the [[Kronecker delta]].

More generally, let <math> (R,\Delta) </math> be a partial [[differential ring]] with commuting derivatives <math> \Delta = \lbrace \partial_1,\ldots,\partial_m \rbrace </math>. The '''Weyl algebra associated to''' <math>(R,\Delta)</math> is the noncommutative ring <math> R[\partial_1,\ldots,\partial_m] </math> satisfying the relations <math> \partial_i r = r\partial_i + \partial_i(r) </math> for all <math> r \in R </math>. The previous case is the special case where <math> R=F[x_1,\ldots,x_n] </math> and <math> \Delta = \lbrace \partial_{x_1},\ldots,\partial_{x_n} \rbrace </math> where <math> F </math> is a field.

This article discusses only the case of <math>A_n</math> with underlying field <math>F</math> [[Characteristic (algebra)|characteristic zero]], unless otherwise stated.

The Weyl algebra is an example of a [[simple ring]] that is not a [[matrix ring]] over a [[division ring]]. It is also a noncommutative example of a [[domain (ring theory)|domain]], and an example of an [[Ore extension]].