7
$\begingroup$

In general there seems no way to naturally define the tensor product of two $A_\infty$ algebras $A$ and $B$. But, if $(A, m^A_1,m^A_2)$ is only a DGA(differential graded algebra) and $(B, m^B_k, k\ge 1) $ is an $A_\infty$ algebra, then is there a natural way to get an $A_\infty$ algebra structure on the tensor product $A\otimes B$?

I guess this should be correct. But for the safety, I was wondering if there is a standard reference for this fact.

Moreover, if this is right, what I really want is an explicit formula of the $A_\infty$ algebra structure on $A\times B$ in terms of $m^A$ and $m^B$. Thank you!

$\endgroup$

1 Answer 1

11
$\begingroup$

In fact the tensor product of two $A_\infty$ algebras can be made into an $A_\infty$ algebra in an explicit way: there are two constructions, one by Saneblidze-Umble and one by Loday. See the paper https://arxiv.org/abs/0710.0572

(For cofibrancy reasons one also knows abstractly that there is such a tensor product, but this of course doesn't give a formula.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.