I am wondering if there is a name for an algebra which is commutative up to some group action.
To be more concrete, assume $A= \bigoplus A_n$ is a graded algebra, so $A_n \cdot A_m \subset A_{n+m}$, and each of the pieces $A_n$ comes equipped with an action of the symmetric group $S_n$ such that $$ a \cdot b = \sigma * (b \cdot a) $$ for some $\sigma \in S_n$ (which depends on $a$ and $b$). The tensor algebra or the exterior algebra of a finite-dimensional vector space are obvious examples of such things.
Is there a name for such a property?