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Abstract
Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod–Shafarevichtype theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider twogenerated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree
n⩾3 is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class Pn of potential algebras with homogeneous potential of degree n+1⩾4, the minimal Hilbert series is Hn=1/1−2t+2t^{n}−t^{n+1}
, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.
n⩾3 is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class Pn of potential algebras with homogeneous potential of degree n+1⩾4, the minimal Hilbert series is Hn=1/1−2t+2t^{n}−t^{n+1}
, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.
Original language  English 

Number of pages  23 
Journal  International Mathematics Research Notices 
DOIs  
Publication status  Published  12 Jan 2018 
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Agata Smoktunowicz
 School of Mathematics  Personal Chair in Algebra
Person: Academic: Research Active