Description of noncommutative symplectic manifolds for self-injective Nakayama algebras
Volume 73 / 2025
Bulletin Polish Acad. Sci. Math. 73 (2025), 117-139
MSC: Primary 14A22; Secondary 58B34, 83C65
DOI: 10.4064/ba250630-20-1
Published online: 29 January 2026
Abstract
We determine all possible exact noncommutative symplectic structures for certain path algebras with relations. These algebras are the path algebras of a cyclic quiver with $r+1$ arrows quotiented by the ideal generated by the paths of length $l$. The main result is that there are exact noncommutative symplectic structures only when $l=s(r+1)+1$ with $s\geq 1$. In this case a description of the open subset of one-forms $z$ such that $dz$ is a non-degenerate symplectic form is given, by reducing it to the case $s=1$.
