A classification of n-representation infinite algebras of type $$\tilde{A}$$

Abstract We classify n -representation infinite algebras $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> of type $$\tilde{A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> . This type is defined by requiring that $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> has higher preprojective algebra $$\Pi _{n+1}(\Lambda ) \simeq k[x_1, \ldots , x_{n+1}] *G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Π</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≃</mml:mo> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> , where $$G \le {{\,\textrm{SL}\,}}_{n+1}(k)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>≤</mml:mo> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>SL</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is finite abelian. For the classification, we group these algebras according to a more refined type, and give a combinatorial characterisation of these types. This is based on so-called height functions, which generalise the height function of a perfect matching in a dimer model. In terms of toric geometry and McKay correspondence, the types form the lattice simplex of junior elements of G . We show that all algebras of the same type are related by iterated n -APR tilting, and hence are derived equivalent. By disallowing certain tilts, we turn this set into a finite distributive lattice, and we construct its maximal and minimal elements.