# Derived equivalent algebras

Given a finite dimensional connected quiver algebra A, define $S_A$ as the set of quiver algebras derived equivalent to $A$ (up to isomorphism). Questions:

1. Can one characterise algebras $A$,where $S_A$ is a finite set?
2. Can one characterise algebras $A$, where $S_A$ has cardinality 1? (For example local algebras)
3. Can one compute $S_A$ when $A$ is the Nakayama algebra with Kupisch series [2,3]?
4. Can one give examples of algebras $A_n$ with $S_{A_n}$ having cardinality $n$?
• For question 4, do you want $A_n$ to be connected? Otherwise you could take any algebra $A$ with $S_A=2$ and let $A_n$ be the direct product of $n-1$ copies of $A$. – Jeremy Rickard Jan 2 '17 at 10:17
• yes, A should be conencted. I edited it. – Mare Jan 2 '17 at 11:52