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Concept
Algebra representation
Summary
In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra.
Linear complex structure
One of the simplest non-trivial examples is a linear complex structure, which is a representation of the complex numbers C, thought of as an associative algebra over the real numbers R. This algebra is realized concretely as which corresponds to i2 = −1. Then a representation of C is a real vector space V, together with an action of C on V (a map ). Concretely, this is just an action of i , as this generates the algebra, and the operator representing i (the of i in End(V)) is denoted J to avoid confusion with the identity matrix I.
Another important basic class of examples are representations of polynomial algebras, 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 k variables over the field K is concretely a K-vector space with k commuting operators, and is often denoted meaning the representation of the abstract algebra where
A basic result about such representations is that, over an algebraically closed field, the representing matrices are simultaneously triangularisable.
Even the case of representations of the polynomial algebra in a single variable are of interest – this is denoted by and is used in understanding the structure of a single linear operator on a finite-dimensional vector space. Specifically, applying the structure theorem for finitely generated modules over a principal ideal domain to this algebra yields as corollaries the various canonical forms of matrices, such as Jordan canonical form.
Official source
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