Algebra representation

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.

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 ${\displaystyle \mathbb {C} =\mathbb {R} [i]/(i^{2}+1),}$ 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 ${\displaystyle \mathbb {C} \to \mathrm {End} (V)}$). Concretely, this is just an action of i , as this generates the algebra, and the operator representing i (the image of i in End(V)) is denoted J to avoid confusion with the identity matrix I).

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