Editing Algebra representation (section)

== Examples ==
=== Linear complex structure ===
{{main|Linear complex structure}}
One of the simplest non-trivial examples is a [[linear complex structure]], which is a representation of the [[complex number]]s '''C''', thought of as an associative algebra over the [[real number]]s '''R'''. This algebra is realized concretely as <math>\mathbb{C} = \mathbb{R}[x]/(x^2+1),</math> which corresponds to {{math|1={{mvar|i}}<sup>2</sup> = −1}}. Then a representation of '''C''' is a real [[vector space]] ''V'', together with an action of '''C''' on ''V'' (a map <math>\mathbb{C} \to \mathrm{End}(V)</math>). Concretely, this is just an action of {{mvar|i}} , as this generates the algebra, and the operator representing {{mvar|i}} (the [[Image_(mathematics)#Image_of_an_element|image]] of {{mvar|i}} in End(''V'')) is denoted ''J'' to avoid confusion with the [[identity matrix]] ''I''.

=== Polynomial algebras ===
Another important basic class of examples are representations of [[polynomial algebra]]s, the free commutative algebras – these form a central object of study in [[commutative algebra]] and its geometric counterpart, [[algebraic geometry]]. A representation of a polynomial algebra in {{mvar|k}} variables over the [[field (mathematics)|field]] ''K'' is concretely a ''K''-vector space with {{mvar|k}} commuting operators, and is often denoted <math>K[T_1,\dots,T_k],</math> meaning the representation of the abstract algebra <math>K[x_1,\dots,x_k]</math> where <math>x_i \mapsto T_i.</math>

A basic result about such representations is that, over an [[algebraically closed field]], the representing [[matrix (mathematics)|matrices]] are [[Triangular matrix#Simultaneous triangularisability|simultaneously triangularisable]].

Even the case of representations of the polynomial algebra in a single variable are of interest – this is denoted by <math>K[T]</math> and is used in understanding the structure of a single [[linear operator]] on a [[dimension (vector space)|finite-dimensional]] vector space. Specifically, applying the [[structure theorem for finitely generated modules over a principal ideal domain]] to this algebra yields as [[Structure theorem for finitely generated modules over a principal ideal domain#Corollaries|corollaries]] the various canonical forms of matrices, such as [[Jordan canonical form]].

In some approaches to [[noncommutative geometry]], the free noncommutative algebra (polynomials in non-commuting variables) plays a similar role, but the analysis is much more difficult.