nLab Jacobian algebra

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

• Zheng Hua, Guisong Zhou, Noncommutative Mather–Yau theorem and its applications to Calabi–Yau algebras, Math. Ann. (2022) doi; Quasi-homogeneity of potentials, J. Noncommut. Geom. 15 (2021) 399–422 doi
• Zheng Hua, Bernhard Keller, Quivers with analytic potentials, arXiv:1909.13517

It is called derivation-quotient algebra in

• Raf Bocklandt, Travis Schedler, Michael Wemyss, Superpotentials and higher order derivations, J. Pure and Appl. Alg. 214:9 (2010) 1501-1522 doi

Other literature

• Daniel Labardini-Fragoso, Bea Schumann, Landau-Ginzburg potentials via projective representations, arXiv:2208.00028

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

category: algebra