Describing the algebra representations of the Laurent polynomials Let $k$ be your favorite field. The "Laurent polynomials" in one variable is the commutative polynomial algebra $k[x,x^{-1}]$ in the two variables $x$ and $x^{-1}$ together with the obvious relation $x.x^{-1} = 1$. Where can I find a decsription of the representations of this algebra?
Edit: By a representation I mean a linear map from $k[x,x^{-1}]$ to the matrices on a finite-dim $k$-vector space (where multiplication is composition of linear maps).
 A: To expand on Captain Lama's comments :
$k[x,x^{-1}]$ is canonically isomorphic to the group algebra $k[\mathbb{Z}]$. Indeed, we have a group homomorphism $\mathbb{Z}\mapsto k[x,x^{-1}]$ sending $1$ to $x$, which induces a $k$-algebra morphism $k[\mathbb{Z}]\to k[x,x^{-1}]$, which is easily seen to be an isomorphism.
Now, for any group $G$, the set of $k$-algebra representations $k[G]\to End(V)$ is canonically isomorphic to the set of (finite dimensional) linear representations $G\to GL(V)$.
Indeed, if we have a linear representation $G\to GL(V)\subset End(V)$, it extends to a $k$-algebra morphism $k[G]\to End(V)$ .
Conversely, since $G\subset k[G]^\times$, a  $k$-algebra morphism $k[G]\to End(V)$ restricts to a group morphism $G\to End(V)^\times=GL(V)$. Both constructions are mutally inverse.
Hence for $G=\mathbb{Z}$, it is enough to compute $Hom_{grps}(\mathbb{Z},GL(V))$, which is canonically isomorphic to $GL(V)$, since a morphism from $\mathbb{Z}$ to any group is uniquely determined by the image of $1$.
Note that two representations $V$, $V'$ of $\mathbb{Z}$ will be equivalent if and only if the corresponding automorphisms $u,u'$ are conjugate (by an isomorphism $V\overset{\sim}{\to} V'$).
So, if you keep track of all identifications, you will end up with the following description: all the representations $k[x,x^{-1}]\to End(V)$ have the form $\rho_{u}:k[x,x^{-1}]\to GL(V)$, where $u\in GL(V)$ and $\rho_u(\displaystyle\sum_{m\in\mathbb{Z}} a_m x^m)=\sum_{m\in\mathbb{Z}}a_mu^m$ (where the $a_m'$ s are all zero except for a finite number of them).
Moreover, equivalence  classes of algebra representations corresponds to similitary classes of automorphisms.
