Related papers: Noncommutative versions of some classical biration…
The three dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress and Miller have proved that in the case of non degenerate…
We study the differential smoothness of Sklyanin algebras in three and four variables. We show that all non-degenerate three-dimensional cases are differentially smooth, while none of the four-dimensional Sklyanin algebras admit a connected…
For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and…
We study a multi-parametric family of quadratic algebras in four generators, which includes coordinate algebras of noncommutative four-planes and, as quotient algebras, noncommutative three spheres. Particular subfamilies comprise Sklyanin…
We initiate a study of projections and modules over a noncommutative cylinder, a simple example of a noncompact noncommutative manifold. Since its algebraic structure turns out to have many similarities with the noncommutative torus, one…
In this article a complete description is given of the simple representations of a 3-dimensional Sklyanin algebra associated to a torsion point. In order to determine these irreducible representations, a review is given of classical results…
We construct a class of noncommutative crepant resolutions of any Kleinian singularity in the form of noncommutative algebras over its crepant partial resolutions. We argue that such resolutions are Morita equivalent to the canonical…
We investigate the most general non(anti)commutative geometry in N=1 four-dimensional superspace, invariant under the classical (i.e., undeformed) supertranslation group. We find that a nontrivial non(anti)commutative superspace geometry…
We discuss examples of non-commutative spaces over non-archimedean fields. Those include non-commutative and quantum affinoid algebras, quantized K3 surfaces and quantized locally analytic p-adic groups.
We find the connection between 3-dimensional commutative algebras with a trivial trace and plane quartics and its bitangents.
Noncommutative domain algebras are noncommutative analogues of the algebras of holomorphic functions on domains of $\C^n$ defined by holomorphic polynomials, and they generalize the noncommutative Hardy algebras. We present here a complete…
The 3-dimensional Sklyanin algebras, S(a,b,c), over a field k, form a flat family parametrized by points (a,b,c) lying in P^2-D, the complement of a set D of 12 points in the projective plane, P^2. When (a,b,c) is in D the algebras having…
In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded…
In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is to determine the minimal models within any birational class. In this…
Here we construct a map from the algebra of fields in two-dimensional noncommutative of U(1) Yang-Mills fields interacting with Kaluza-Klein scalars to a D-dimensional one, as a solution in the two-dimensional model. This proves the…
Noncommutative three-dimensional gravity can be described in terms of a noncommutative Chern-Simons theory. We extend this structure and also propose an action for gravitational fields on an even dimensional noncommutative space. The action…
In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of…
We construct a functor from the category of elliptic curves to a category of noncommutative tori. Our proof is based on an isomorphism between the Sklyanin algebras and dense sub-algebras of the noncommutative tori.
We consider the noncommutative space-times with Lie-algebraic noncommutativity (e.g. $\kappa$-deformed Minkowski space). In the framework with classical fields we extend the $\star$-product in order to represent the noncommutative…
Based on an argument for the noncommutativity of momenta in noncommutative directions, we arrive at a generalization of the ${\cal N}=1$ super $E^2$ algebra associated to the deformation of translations in a noncommutative Euclidean plane.…