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.Comment: 27 pages; v2: minor corrections and additional reference