# The endomorphisms of an $A_\infty$-algebra as a (bi)module over itself

Let $A$ be a unital associative algebra. A well-known exercise states that the ring of $A$-bimodule endomorphisms of $A$ are isomorphic to the center of $A$. That is, $\text{End}_{A-\text{mod}}(A) \cong Z(A)$.

I would like to extend this result to the $A_\infty$ world. Let $\mathcal{A}$ be a strictly unital $A_\infty$-algebra. To keep things simple, let's suppose that the differential of $\mathcal{A}$ is trivial, that $\mathcal{A}$ is strictly unital, and that the characteristic of the underlying field is 2. (Maybe this last bit isn't a simplification?) Write $e$ for the unit of $\mathcal{A}$.

Let $\mathcal{E} = \text{End}_{\mathcal{A}-\text{mod}}(\mathcal{A})$ denote the endomorphisms of $\mathcal{A}$ as a strictly unital $\mathcal{A}$-bimodule. An element $f \in \mathcal{E}$ is a sequence of maps $$f_{i,j} : \mathcal{A}^{\otimes i} \otimes \mathcal{A} \otimes \mathcal{A}^{\otimes j} \to \mathcal{A}$$ which obey certain associativity rules. These rules imply that $f$ is determined by its values on tensors of the form $$x_1 \otimes \cdots \otimes x_i \otimes e \otimes y_1 \otimes \cdots \otimes y_j,$$ i.e. simple tensors with the unit in the 'module slot.' They also imply that $f_1(e)$ is in the center of $\mathcal{A}$.

I am asking if a complete characterization, maybe up to homotopy, is known. There are some hints of this in Appendix B of Lefèvre-Hasegawa's thesis involving Hochschild cohomology. But I don't know if obstruction theory is helpful here. I appreciate any help, references, or indications that this is hopeless!

• The good notion should be up to equivalence, and then you can replace the algebra by a real DGA, no? – Mariano Suárez-Álvarez Jun 14 '16 at 23:01
• @MarianoSuárez-Alvarez Thank you! I'm new to this, so let me try to answer your rhetorical question: Every A-infinity algebra is quasi-isomorphic to a DGA. A quasi-isomorphism of A-infinity algebras (or modules) is a homotopy equivalence. So up to homotopy, we can just study the endomorphisms of this DGA as a module over itself, which are classified by the center. – Adam Saltz Jun 14 '16 at 23:43
• There is such a theorem in Lurie's Higher Algebra, but the setting is $\infty$-categories, not dg-modules as you are considering (thus things are rather more complicated). There's probably a way to explicitly see the correspondence in your setting. – Najib Idrissi Jun 15 '16 at 9:30