4,309 research outputs found
Automorphism groupoids in noncommutative projective geometry
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 is isomorphic to
commutative . 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
Examples of noncommutative manifolds: complex tori and spherical manifolds
We survey some aspects of the theory of noncommutative manifolds focusing on
the noncommutative analogs of two-dimensional tori and low-dimensional spheres.
We are particularly interested in those aspects of the theory that link the
differential geometry and the algebraic geometry of these spaces.Comment: Survey article. Final version. To appear in the proceedings volume of
the "International Workshop on Noncommutative Geometry", IPM, Tehran 200
Quasideterminants
The determinant is a main organizing tool in commutative linear algebra. In
this review we present a theory of the quasideterminants defined for matrices
over a division algebra. We believe that the notion of quasideterminants should
be one of main organizing tools in noncommutative algebra giving them the same
role determinants play in commutative algebra.Comment: amstex; final version; to appear in Advances in Mat
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