Timeline for Confusion about signs in the definition of an $A_\infty$-algebra
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 20 at 20:14 | answer | added | cbz20 | timeline score: 2 | |
| Nov 24, 2023 at 11:20 | comment | added | Igor Khavkine | To make unwinding this definition even more intuitive, the axioms satisfied by $d$ are equivalent to its adjoint $d^*$ being a square zero, degree +1 derivation on the tensor algebra $T(V^*[-1])$. I'm assuming here that the grading on $V^*$ is defined such that the natural duality paring between $T(V[1])$ and $T(V^*[-1])$ itself has degree zero. To give oneself extra peace of mind, one can work out all the formulas for the $m_k$ in the case when $V$ is finite dimensional (or at least degree-wise finite dimensional), and then declare them the correct axioms for $A_\infty$-algebra for any $V$. | |
| Nov 24, 2023 at 3:02 | comment | added | Denis T | The right definition of a (strictly unital, augmented) $A_{\infty}$-algebra structure on a graded space $V \oplus 1$ is a square zero degree -1 coderivation on the tensor coalgebra $T(V[1])$. (One may argue that there are other equivalent definitions; but I'd say that you cannot make this one meaningless by one sign error, unlike those with sums over partitions). If you unwind all graded pieces of the equation $d^2 = 0$ carefully enough, Koszul rule will tell you everything you need about the signs. | |
| S Nov 23, 2023 at 23:43 | review | First questions | |||
| Nov 24, 2023 at 5:33 | |||||
| S Nov 23, 2023 at 23:43 | history | asked | ainfg | CC BY-SA 4.0 |