An algebra over the associative operad

Question

So an algebra over the Associative operad is an operad morphisme $Ass \rightarrow End_V $. And is an associative algebra. Are there any nice examples of an associative algebras definined this way? I have not been able to find anything on this.

Answer 1

For any associative algebra $V$ you have a map $m:V\otimes V\to V$ which is precisely the associative multiplication. This is an element $m\in End_V(2)$. The operad $Ass$ is generated by an associative binary operation $\mu\in Ass(2)$ so an operad morphism $Ass\to End_V$ is determined by where you send $\mu$, so you just have to define $\mu\to m$.

So for any example you do need to specify what the map $m$ is if you want to make it explicit. For instance, there are several associative operations on the real numbers (the usual multiplication and sum, for example), so each of them define a morphism from the assocciative operad to $End_{\mathbb{R}}$.

Answer 2

Actually, and this is the marvelous thing about operads, $\textit{any}$ associative algebra arises in this way. Take your favorite vector space $A$, with your favorite associative algebra structure, and there $\textit{is}$ a morphism $\mathcal{Ass}\to\textrm{End}_A$, defined as in Javi's answer. So the associative operad is "universally" encoding the structure of an associative algebra, capturing its "essence".