We construct all possible noncommutative deformations of a Kleinian singularity ${\mathbb C}^2/Γ$ of type $D_n$ in terms of generators and relations, and solve the problem of when two deformations are isomorphic. We prove that all isomorphisms arise naturally from the action of the normalizer $N_{\SL(2)}(Γ)$ on ${\mathbb C}/Γ$. We deduce that the moduli space of isomorphism classes of noncommutative deformations in type $D_n$ is isomorphic to a vector space of dimension $n$.
We establish a version of Knörrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let $A$ be a left noetherian AS-regular algebra, let $f$ be a normal and regular element of $A$ of positive degree, and take $B=A/(f)$. Then there exists a bijection between the set of isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules over $B$ and those over (a noncommutative analog of) its second double branched cover $(B^\#)^\#$. Our results use and extend the study of twisted matrix factorizations, which was introduced by the first three authors with…
For $n\geq 4$ we shall construct a family $D(q)$ of non-commutative deformations of the coordinate algebra of a Kleinian singularity of type $D_n$ depending on a polynomial $q$ of degree $n$. We shall prove that every deformation of a type $D$ Kleinian singularity which is not commutative is isomorphic to some $D(q)$. We shall then consider in type $D$ the family of deformations $\mathcal{O}^{\boldsymbolλ}$ constructed by Crawley-Boevey and Holland. For each $\mathcal{O}^{\boldsymbolλ}$ which is not commutative we shall exhibit an explicit isomorphism $D(q)\cong \mathcal{O}^{\boldsymbolλ}$ for…
Let $G$ be a finite subgroup of $\text{SL}(2,\Bbbk)$ and let $R = \Bbbk[x,y]^G$ be the coordinate ring of the corresponding Kleinian singularity. In 1998, Crawley-Boevey and Holland defined deformations $\mathcal{O}^λ$ of $R$ parametrised by weights $λ$. In this paper, we determine the singularity categories $\mathcal{D}_{\text{sg}}(\mathcal{O}^λ)$ of these deformations, and show that they correspond to subgraphs of the Dynkin graph associated to $R$. This generalises known results on the structure of $\mathcal{D}_{\text{sg}}(R)$. We also provide a generalisation of the intersection theory app…
In noncommutative algebraic geometry, noncommutative quadric hypersurfaces are major objects of study. In this paper, we focus on studying noncommutative conics $\operatorname{Proj_{nc}} A$ embedded into Calabi-Yau quantum projective planes. In particular, we give complete classifications of homogeneous coordinate algebras $A$ of noncommutative conics up to isomorphism of graded algebras, and of noncommutive conics $\operatorname{Proj_{nc}} A$ up to isomorphism of noncommutative schemes.
For a noncommutative space X, we study Inj(X), the set of isomorphism classes of indecomposable injective X-modules. In particular, we look at how this set, suitably topologized, can be viewed as an underlying "spectrum" for X. As applications we discuss noncommutative notions of irreducibility and integrality, and a way of associating an integral subspace of X to each element of Inj(X) which behaves like a "weak point."
Let $A$ be a Koszul Calabi-Yau algebra. We show that there exists an isomorphism of Batalin-Vilkovisky algebras between the Hochschild cohomology ring of $A$ and that of its Koszul dual algebra $A^!$. This confirms (a generalization of) a conjecture of R.~Rouquier.
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 define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P.M. Soltan. Also, algebraic families of (all) isomorphisms are introduced. By using these notions we construct two classes of Hopf-algebras which may be interpreted as the quantum group of all maps from a finite space to a quantum group, and the quantum group of all automorphisms of a finite NC space. As special cases three classes of NC objects are introduced: quantum group of gauge transformations, Pontryagin dual of a quantum group, and Galois-Hopf-algebra of a…
Recently the Euler forms on numerical Grothendieck groups of rank 4 whose properties mimick that of the Euler form of a smooth projective surface have been classified. This classification depends on a natural number $m$, and suggests the existence of noncommutative surfaces which up to that point had not been considered for $m\geq 2$. These have been constructed for $m=2$ using noncommutative $\mathbb{P}^1$-bundles, and for all $m\geq 2$ by a different construction using maximal orders on $\mathrm{Bl}_x\mathbb{P}^2$. In this article we compare the constructions for $m=2$, i.e. we compare the c…
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 alg…
This paper gives an elementary introduction to noncommutative deformations of modules. The main results of this deformation theory are due to Laudal. Let k be an algebraically closed (commutative) field, let A be an associative k-algebra, and let M = {M_1, ..., M_p} be a finite family of left A-modules. We study the simultaneous formal deformations of the family M, described by the noncommutative deformation functor Def(M): a(p) -> Sets introduced by Laudal. In particular, we prove that the deformation functor Def(M) has a pro-representing hull H(M), unique up to non-canonical isomorphism, and…
We discuss a possible noncommutative generalization of the notion of an equivariant vector bundle. Let $A$ be a $\mathbb{K}$-algebra, $M$ a left $A$-module, $H$ a Hopf $\mathbb{K}$-algebra, $δ:A\to H\otimes A:=H\otimes_{\mathbb{K}} A$ an algebra coaction, and let $(H\otimes A)_δ$ denote $H\otimes A$ with the right $A$-module structure induced by~$δ$. The usual definitions of an equivariant vector bundle naturally lead, in the context of $\mathbb{K}$-algebras, to an $(H\otimes A)$-module homomorphism \[Θ:H\otimes M\to (H\otimes A)_δ\otimes_AM\] that fulfills some appropriate conditions. On the …
During the last decades algebraization of space turned out to be a promising tool at the interface between Mathematics and Theoretical Physics. Starting with works by Gel'fand-Kolmogoroff and Gel'fand-Naimark, this branch developed as from the fortieth in two directions: algebraic characterization of usual geometric space on the one hand, and algebraically defined noncommutative space, which is known to be tightly related with e.g. quantum gravity and super string theory, on the other hand. In this note, we combine both aspects, prove a superversion of Shanks and Pursell's classical result sta…
