Definition 1

Let M be a K-module. A linear representation of the algebra A in M is a K-homomorphism $\pi$ from A to the algebra End$_K(M)$.

We also say that the pair $(M, \pi )$ is a linear representation of the algebra A. When M is a free K-module, the dimension of M is called the degree (or dimension) of $\pi$.

Let $\pi$ be a linear representation of A in M. The additive law on M and the external law $(a, x)\mapsto \pi (a)(x)$ define a left A-module structure on M, which we denote by $M_{\pi}$. We say that $M_{\pi}$ is the module of the representation $\pi$. The K-module structure on M is obtained from the A-module structure on $M_{\pi}$ by restriction of scalars.

Conversely, let E be a left A-module. Let M be the K-module deduced from E by restriction of scalars from A to K, and let $\pi$ be the homomorphism $a\mapsto a_E$ from A to End$_K(M)$. Then $\pi$ is a representation of A in M, and we have $E = M_{\pi}$. We say that $\pi$ is the linear representation associated with the A-module E.

Studying A-modules or linear representations of A amounts to the same. We will translate certain definitions concerning modules to the language of linear representations.

Let $\pi$ be a linear representation of A in M. The kernel of the homomorphism $\pi$ is a two-sided ideal of A, which is simply the annihilator of the A-module $M_{\pi}$. The homomorphism $\pi$ is injective if and only if the A-module $M_{\pi}$ is faithful; we then say that $\pi$ is a faithful representation of A.

Let $(M, \pi )$ and $(M', \pi ')$ be linear representations of A. An A-linear mapping from $M_{\pi}$ to $M'_{\pi'}$ is a K-linear mapping $u: M\rightarrow M'$ that satisfies the relation

for $a\in A$. An isomorphism from $M_{\pi}$ to $M'_{\pi'}$ is an isomorphism of K-modules $u: M\rightarrow M'$ satisfying the condition

for $a\in A$. We say that $\pi$ and $\pi '$ are isomorphic if the A-modules $M_{\pi}$ and $M'_{\pi'}$ are isomorphic. We say that $\pi$ is a subrepresentation (resp. a quotient representation) of $\pi '$ if $M_{\pi}$ is a submodule (resp. a quotient module) of $M'_{\pi'}$.

Suppose given a family $(M_i, \pi_i)$ of linear representations of A. Let M be the K-module that is the direct sum of the $M_i$; for any $a\in A$, let $\pi (a)$ be the endomorphism $(x_i)_{i\in I}\mapsto (\pi_i(a)(x_i))_{i\in I}$ of M. Then $\pi$ is a linear representation of A in M. The A-module $M_{\pi}$ is the direct sum of the A-modules $(M_i)_{\pi}(i\in I)$; we say that $\pi$ is the direct sum of the $\pi_i$, and we write $\pi =\bigoplus\pi_{i^i}$.

We say that the representation $\pi$ of A in M is simple or irreducible if the A-module $M_{\pi}$ is simple; we say that the representation $\pi$ of A on M is semisimple or completely reducible if the A-module $M_{\pi}$ is semisimple.

Let $\pi$ be a linear representation of A in M. Let $M^*$ be the K-module dual to M. The mapping $a\mapsto^t(\pi (a))$ from $A^o$ to End$_K(M^*)$ is the transposed representation of $\pi$.

Let L be a commutative K-algebra, and let $(M, \pi )$ be a linear representation of the K-algebra A. The homomorphism $\pi_{(L)}: A_{(L)}\rightarrow$ End$_L(M_{(L)})$ corresponding to the $A_{(L)}$-module structure on $M_{(L)}$ is a linear representation of the L-algebra $A_{(L)}$. We say that $\pi_{(L)}$ is the linear representation of the algebra $A_{(L)}$ deduced from the representation $\pi$ by extending the ring of scalars K to L.

Suppose that K is a field and that L is a nonzero commutative K-algebra.

Let $\pi$ and $\pi '$ be linear representations of the algebra A. It follows from VIII, p. 37, Theorem 3 that the representations $\pi$ and $\pi '$ are isomorphic if and only if $\pi_{(L)}$ and $\pi '_{(L)}$ are.

Suppose that K is a field. Let L be an extension of K. Consider the Grothendieck group $R_K(A)$ (resp. $R_L(A_{(L)})$) of the A-modules that are finite-dimensional over K (resp. of the $A_{(L)}$-modules that are finite-dimensional over L). We have seen that the group homomorphism

defined by extension of scalars is injective; moreover, an element $\xi \in R_K(A)$ is effective if and only if $u(\xi )$ is (VIII, p. 195, Theorem 1).