Definition:Linear Representation/Algebra

Definition

Let $K$ be a field.

Let $A$ be an associative unitary algebra over $K$.

Then a (linear) representation of $A$ is a vector space $V$ over $K$ equipped with a homomorphism of algebras:

$\rho: A \to \map {\operatorname {End} } V$

where $\map {\operatorname {End} } V$ is the endomorphism ring of $V$.