Higher representation infinite algebras and toric Fano stacks of Picard number one or two

Abstract

Tilting bundles translate geometry into non-commutative algebra via derived equivalences. We prove the existence of, and classify, $d$-tilting bundles consisting of line bundles on $d$-dimensional smooth toric Fano stacks of Picard number one or two. Their endomorphism algebras give natural examples of $d$-representation infinite algebras and are closely related to the derived McKay correspondence. The classification is motivated by dimer models: an internal perfect matching gives a positive grading on a dimer algebra, whose degree-zero part yields a $2$-representation infinite algebra. The algebras of type $\widetilde A$ introduced by Herschend--Iyama--Oppermann are higher-dimensional analogues of this construction in the simplex case. Applying the same principle to the next case leads to a new class of higher representation infinite algebras, which we call algebras of type $\widetilde A\widetilde A$. Upper sets provide a common framework for tilting bundles, toric non-commutative crepant resolutions (NCCRs), and cuts of higher-dimensional dimer-type quivers. In the Picard-number-one case, $d$-tilting bundles consisting of line bundles are parametrized by non-trivial upper sets in the Picard group, and their endomorphism algebras are precisely the algebras of type $\widetilde A$. In the Picard-number-two case, the upper-set construction becomes two-step: the first upper set determines the ambient toric NCCR, and the second selects an internal cut of its quiver. This classifies all such $d$-tilting bundles and realizes their endomorphism algebras precisely as algebras of type $\widetilde A\widetilde A$. Thus smooth toric Fano stacks of Picard number one and two serve as geometric models of algebras of type $\widetilde A$ and $\widetilde A\widetilde A$, respectively. Using these models, we also show that both classes are closed under $d$-APR tilts.