Left-right non-bimodule examples Let $A$ be a unital algebra, defined over the complex numbers. Any bimodule $M$ over $A$ must, by definition, be a left, and right, module satisfing
$$
a.(m.b) = (a.m).b, ~~~~~~~ \textrm{ for all } a,b \in A, ~ m \in M
$$
What is a "natural" or "well-motivated" example of a an object which is both a left and right module, but does not satisfy the above condition?
 A: Suppose $A$ is a non-commutative Hopf algebra. Then you can use $M=A$ with the left adjoint and right regular actions:
$$
a\cdot m = \sum_{(a)}a_{(1)}mS(a_{(2)}), \ m\cdot b = mb .
$$
In particular, you can use the group algebra of a a group $G$ so that
$$
g\cdot m = gmg^{-1}, \ m\cdot h = mh, \ g,h,m\in G.
$$
A: $\newcommand{\C}{\mathbb C}$
Let $G$ be a nonabelian group and $\rho\colon G\to\mathrm{Aut}(V)$ be a faithful representation of $G$ over $\C$.
Then $V$ is naturally a left $\C[G]$-module via the action given by $g\cdot v := \rho(g)(v)$, and is naturally a
right $\C[G]$-module via the action given by $v\cdot g := \rho(g^{-1})(v)$. (Here we use the fact that $\C[G]$ is
generated as an algebra by the elements of $G$, so one can define a module action using the elements of $G$ and
extend to a $\C[G]$-action.)
If $V$ is a bimodule, then for all $g,h\in G$, $g\cdot (v\cdot h^{-1}) = (g\cdot v)\cdot h^{-1}$, but the left-hand
side is $\rho(gh)v$ and the right-hand side is $\rho(hg)v$. But $\rho$ is faithful and $G$ is nonabelian, so this
cannot occur.
