Given a quiver $Q$ with potential which is an element in a (possibly completed) path algebra ${\mathbf C} Q$ of $Q$, one can consider cyclic derivations $\partial_i:HH_0({\mathbf C} Q)\to {\mathbf C} Q$ and the “Jacobian” ideal in ${\mathbf C} Q$ generated by all of them. The quotient algebra is called Jacobi algebra or a Jacobian algebra of the quiver. It is a noncommutative analogue of the Milnor algebra of a hypersurface singularity?.
Its role in noncommutative algebraic geometry analoguous to describing some properties of hypersurface singularities is partial achieved in
It is called derivation-quotient algebra in
Other literature
We interpret the Landau-Ginzburg potentials associated to Gross-Hacking-Keel-Kontsevich’s partial compactifications of cluster varieties as F-polynomials of projective representations of Jacobian algebras. Along the way, we show that both the projective and the injective representations of Jacobi-finite quivers with potential are well-behaved under Derksen-Weyman-Zelevinsky’s mutations of representations.
C. Geiß, D. Labardini-Fragoso, J. Schröer, The representation type of Jacobian algebras, Adv. Math. 290 (2016) 364–452
M. Pressland, Mutation of frozen Jacobian algebras, J. Algebra 546 (2020) 236–273
Last revised on June 23, 2023 at 08:05:25. See the history of this page for a list of all contributions to it.