Editing Associative algebra (section)

=== Mathematical physics ===
* A [[Poisson algebra]] is a commutative associative algebra over a field together with a structure of a [[Lie algebra]] so that the Lie bracket {{mset|,}} satisfies the Leibniz rule; i.e., {{nowrap|1={{mset|''fg'', ''h''}} = {{itco|''f''}}{{mset|''g'', ''h''}} + ''g''{{mset|''f'', ''h''}}}}.
* Given a Poisson algebra <math>\mathfrak a</math>, consider the vector space <math>\mathfrak{a}[\![u]\!]</math> of [[formal power series]] over <math>\mathfrak{a}</math>.  If  <math>\mathfrak{a}[\![u]\!]</math> has a structure of an associative algebra with multiplication <math>*</math> such that, for <math>f, g \in \mathfrak{a}</math>,
*: <math>f * g = f g - \frac{1}{2} \{ f, g \} u + \cdots,</math>
: then <math>\mathfrak{a}[\![u]\!]</math> is called a [[deformation quantization]] of <math>\mathfrak a</math>.
* A [[quantized enveloping algebra]]. The dual of such an algebra turns out to be an associative algebra (see {{slink||Dual of an associative algebra}}) and is, philosophically speaking, the (quantized) coordinate ring of a [[quantum group]].
* [[Gerstenhaber algebra]]