题目：Quantum projective planes finite over their centers and Beilinson algebras

报告人：Ayako Itaba (Tokyo University of Science)

时间：2023年12月12日下午2:30-3:20

地点：东区五教5205

摘要：In noncommutative algebraic geometry, a quantum polynomial algebra defined by Artin-Schelter is a basic and important research object, which is a noncommutative analogue of a commutative polynomial algebra. Also, Artin-Schelter gave the classifications of low dimensional quantum polynomial algebras. Moreover, Artin-Tate-Van den Bergh found a nice correspondence between 3-dimensional quantum polynomial algebras and geometric pair (E,σ), where E is the projective plane or a cubic divisor in he projective plane, and σ is the automorphism of E. So, this result allows us to write a 3-dimensional quantum polynomial algebra A as the form A = A(E, σ). 

   For a 3-dimensional quantum polynomial algebra A = A(E,σ), Artin-Tate-Van den Bergh showed that A is finite over its center if and only if the order |σ| of σ is finite. For a 3-dimensional quantum polynomial algebra A = A(E, σ) with the Nakayama automorphism ν of A, the author and Mori proved that the order |ν∗σ3| of ν∗σ3 is finite if and only if the norm ||σ|| of σ introduced by Mori is finite if and only if the noncommutative projective plane in the sense Artin and Zhang is finite over its center. 

   For a 3-dimensional quantum polynomial algebra A of Type S’, the author prove that the following conditions are equivalent; (1) the noncommutative projective plane is finite over its center. (2) The Beilinson algebra of A is 2-representation tame in the sense of Herschend-Iyama-Oppermann. (3) The isomorphism classes of simple 2-regular modules over the Beilinson algebra of A are parametrized by the projective plane. Note that, this result holds for a Type S by Mori. The Beilinson algebra of A is introduced by Minamoto-Mori, which is a typical example of 2-representation infinite algebra.