Editing Associative algebra (section)

== Examples ==

The most basic example is a ring itself; it is an algebra over its [[Center (ring theory)|center]] or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics.

=== Algebra ===
* Any ring ''A'' can be considered as a '''Z'''-algebra. The unique ring homomorphism from '''Z''' to ''A'' is determined by the fact that it must send 1 to the identity in ''A''. Therefore, rings and '''Z'''-algebras are equivalent concepts, in the same way that [[abelian group]]s and '''Z'''-modules are equivalent.
* Any ring of [[characteristic (algebra)|characteristic]] ''n'' is a ('''Z'''/''n'''''Z''')-algebra in the same way.
* Given an ''R''-module ''M'', the [[endomorphism ring]] of ''M'', denoted End<sub>''R''</sub>(''M'') is an ''R''-algebra by defining {{nowrap|1=(''r''·''φ'')(''x'') = ''r''·''φ''(''x'')}}.
* Any ring of [[matrix (mathematics)|matrices]] with coefficients in a commutative ring ''R'' forms an ''R''-algebra under matrix addition and multiplication. This coincides with the previous example when ''M'' is a finitely-generated, [[free module|free]] ''R''-module.
** In particular, the square ''n''-by-''n'' [[square matrix|matrices]] with entries from the field ''K'' form an associative algebra over ''K''.
* The [[complex number]]s form a 2-dimensional commutative algebra over the [[real number]]s.
* The [[quaternion]]s form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions).
* Every [[polynomial ring]] {{nowrap|''R''[''x''<sub>1</sub>, ..., ''x<sub>n</sub>'']}} is a commutative ''R''-algebra. In fact, this is the free commutative ''R''-algebra on the set {{nowrap|{{mset|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}}}}.
* The [[free algebra|free ''R''-algebra]] on a set ''E'' is an algebra of "polynomials" with coefficients in ''R'' and noncommuting indeterminates taken from the set ''E''.
* The [[tensor algebra]] of an ''R''-module is naturally an associative ''R''-algebra. The same is true for quotients such as the [[exterior algebra|exterior]] and [[symmetric algebra]]s. Categorically speaking, the [[functor]] that maps an ''R''-module to its tensor algebra is [[left adjoint]] to the functor that sends an ''R''-algebra to its underlying ''R''-module (forgetting the multiplicative structure).
* Given a module ''M'' over a commutative ring ''R'', the direct sum of modules {{nowrap|1=''R'' ⊕ ''M''}} has a structure of an ''R''-algebra by thinking ''M'' consists of infinitesimal elements; i.e., the multiplication is given as {{nowrap|1=(''a'' + ''x'')(''b'' + ''y'') = ''ab'' + ''ay'' + ''bx''}}. The notion is sometimes called the [[algebra of dual numbers]].
* A [[quasi-free algebra]], introduced by Cuntz and Quillen, is a sort of generalization of a free algebra and a semisimple algebra over an algebraically closed field.

=== Representation theory ===
* The [[universal enveloping algebra]] of a Lie algebra is an associative algebra that can be used to study the given Lie algebra.
* If ''G'' is a group and ''R'' is a commutative ring, the set of all functions from ''G'' to ''R'' with finite support form an ''R''-algebra with the convolution as multiplication. It is called the [[group ring|group algebra]] of ''G''. The construction is the starting point for the application to the study of (discrete) groups.
* If ''G'' is an [[algebraic group]] (e.g., semisimple [[complex Lie group]]), then the [[coordinate ring]] of ''G'' is the [[Hopf algebra]] ''A'' corresponding to ''G''. Many structures of ''G'' translate to those of ''A''.
* A [[quiver algebra]] (or a path algebra) of a directed graph is the free associative algebra over a field generated by the paths in the graph.

=== Analysis ===
* Given any [[Banach space]] ''X'', the [[continuous function (topology)|continuous]] [[linear operator]]s {{nowrap|''A'' : ''X'' → ''X''}} form an associative algebra (using composition of operators as multiplication); this is a [[Banach algebra]].
* Given any [[topology|topological space]] ''X'', the continuous real- or complex-valued functions on ''X'' form a real or complex associative algebra; here the functions are added and multiplied pointwise.
* The set of [[semimartingale]]s defined on the [[filtration (mathematics)#Measure theory|filtered probability space]] {{nowrap|(Ω, ''F'', (''F''<sub>''t''</sub>)<sub>''t''≥0</sub>, P)}} forms a ring under [[stochastic calculus|stochastic integration]].{{citation needed|date=October 2023}}
* The [[Weyl algebra]]
* An [[Azumaya algebra]]

=== Geometry and combinatorics ===
* The [[Clifford algebra]]s, which are useful in [[geometry]] and [[physics]].
* [[Incidence algebra]]s of [[locally finite poset|locally finite]] [[partially ordered set]]s are associative algebras considered in [[combinatorics]].
* The [[partition algebra]] and its subalgebras, including the [[Brauer algebra]] and the [[Temperley-Lieb algebra]].
* A [[differential graded algebra]] is an associative algebra together with a grading and a differential. For example, the [[de Rham algebra]] <math display="inline">\Omega(M) = \bigoplus_{p=0}^n \Omega^p(M)</math>, where <math display="inline">\Omega^p(M)</math> consists of differential ''p''-forms on a manifold ''M'', is a differential graded algebra.

=== 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]]