TY - JOUR

T1 - Automorphism groupoids in noncommutative projective geometry

AU - Cooney, Nicholas

AU - Grabowski, Jan Edward

PY - 2022/8/15

Y1 - 2022/8/15

N2 - We address a natural question in noncommutative geometry, namely the rigidity observed in many examples, whereby noncommutative spaces (or equivalently their coordinate algebras) have very few automorphisms by comparison with their commutative counterparts. In the framework of noncommutative projective geometry, we define a groupoid whose objects are noncommutative projective spaces of a given dimension and whose morphisms correspond to isomorphisms of these. This groupoid is then a natural generalization of an automorphism group. Using work of Zhang, we may translate this structure to the algebraic side, wherein we consider homogeneous coordinate algebras of noncommutative projective spaces. The morphisms in our groupoid precisely correspond to the existence of a Zhang twist relating the two coordinate algebras. We analyse this automorphism groupoid, showing that in dimension 1 it is connected, so that every noncommutative ℙ1 is isomorphic to commutative ℙ1. For dimension 2 and above, we use the geometry of the point scheme, as introduced by Artin-Tate-Van den Bergh, to relate morphisms in our groupoid to certain automorphisms of the point scheme. We apply our results to two important examples, quantum projective spaces and Sklyanin algebras. In both cases, we are able to use the geometry of the point schemes to fully describe the corresponding component of the automorphism groupoid. This provides a concrete description of the collection of Zhang twists of these algebras.

AB - We address a natural question in noncommutative geometry, namely the rigidity observed in many examples, whereby noncommutative spaces (or equivalently their coordinate algebras) have very few automorphisms by comparison with their commutative counterparts. In the framework of noncommutative projective geometry, we define a groupoid whose objects are noncommutative projective spaces of a given dimension and whose morphisms correspond to isomorphisms of these. This groupoid is then a natural generalization of an automorphism group. Using work of Zhang, we may translate this structure to the algebraic side, wherein we consider homogeneous coordinate algebras of noncommutative projective spaces. The morphisms in our groupoid precisely correspond to the existence of a Zhang twist relating the two coordinate algebras. We analyse this automorphism groupoid, showing that in dimension 1 it is connected, so that every noncommutative ℙ1 is isomorphic to commutative ℙ1. For dimension 2 and above, we use the geometry of the point scheme, as introduced by Artin-Tate-Van den Bergh, to relate morphisms in our groupoid to certain automorphisms of the point scheme. We apply our results to two important examples, quantum projective spaces and Sklyanin algebras. In both cases, we are able to use the geometry of the point schemes to fully describe the corresponding component of the automorphism groupoid. This provides a concrete description of the collection of Zhang twists of these algebras.

KW - Noncommutative projective geometry

KW - Zhang twist

KW - Point scheme

U2 - 10.1016/j.jalgebra.2022.03.045

DO - 10.1016/j.jalgebra.2022.03.045

M3 - Journal article

VL - 604

SP - 296

EP - 323

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -