Noncommutative Projective Geometry Research Articles

Automorphisms of quantum polynomial rings and Drinfeld Hecke algebras

We consider quantum (skew) polynomial rings and observe that their graded automorphisms coincide with those of quantum exterior algebras. This allows us to define a quantum determinant that gives a homomorphism of groups acting on quantum polynomial rings. We use quantum subdeterminants to classify the resulting Drinfeld Hecke algebras for the symmetric group, other infinite families of Coxeter and complex reflection groups, and mystic reflection groups (which satisfy a version of the Shephard–Todd–Chevalley theorem). This direct combinatorial approach replaces the technology of Hochschild cohomology used by Naidu and Witherspoon over fields of characteristic zero and allows us to extend some of their results to fields of arbitrary characteristic and also locate new deformations of skew group algebras.

A Borel–Weil theorem for the quantum Grassmannians

We establish a noncommutative generalisation of the Borel–Weil theorem for the celebrated Heckenberger–Kolb calculi of the quantum Grassmannians. The result is formulated in the framework of quantum principal bundles and noncommutative complex structures, and generalises previous work of a number of authors on quantum projective space. As a direct consequence we get a novel noncommutative differential geometric presentation of the twisted Grassmannian coordinate ring studied in noncommutative projective geometry. A number of applications to the noncommutative Kähler geometry of the quantum Grassmannians are also given.

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, using 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, the two-dimensional quantum projective space 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.

Classification of noncommutative conics associated to symmetric regular superpotentials

Let [Formula: see text] be a [Formula: see text]-dimensional quantum polynomial algebra, and [Formula: see text] a central regular element. The quotient algebra [Formula: see text] is called a noncommutative conic. For a noncommutative conic [Formula: see text], there is a finite-dimensional algebra [Formula: see text] which determines the singularity of [Formula: see text]. In this paper, we mainly focus on a noncommutative conic such that its quadratic dual is commutative, which is equivalent to say, [Formula: see text] is determined by a symmetric regular superpotential. We classify these noncommutative conics up to isomorphism of the pairs [Formula: see text], and calculate the algebras [Formula: see text].

Quantum projective planes finite over their centers

AbstractFor a three-dimensional quantum polynomial algebra $A=\mathcal {A}(E,\sigma )$ , Artin, Tate, and Van den Bergh showed that A is finite over its center if and only if $|\sigma |<\infty $ . Moreover, Artin showed that if A is finite over its center and $E\neq \mathbb P^{2}$ , then A has a fat point module, which plays an important role in noncommutative algebraic geometry; however, the converse is not true in general. In this paper, we will show that if $E\neq \mathbb P^{2}$ , then A has a fat point module if and only if the quantum projective plane ${\sf Proj}_{\text {nc}} A$ is finite over its center in the sense of this paper if and only if $|\nu ^{*}\sigma ^{3}|<\infty $ where $\nu $ is the Nakayama automorphism of A. In particular, we will show that if the second Hessian of E is zero, then A has no fat point module.

Exotic elliptic algebras

The 4-dimensional Sklyanin algebras, over $\mathbb {C}$, $A(E,\tau )$, are constructed from an elliptic curve $E$ and a translation automorphism $\tau$ of $E$. The Klein vierergruppe $\Gamma$ acts as graded algebra automorphisms of $A(E,\tau )$. There is also an action of $\Gamma$ as automorphisms of the matrix algebra $M_2(\mathbb {C})$ making it isomorphic to the regular representation. The main object of study in this paper is the invariant subalgebra $\widetilde {A}:=\big (A(E,\tau ) \otimes M_2(\mathbb {C})\big )^{\Gamma }$. Like $A(E,\tau )$, $\widetilde {A}$ is noetherian, generated by 4 degree-one elements modulo six quadratic relations, Koszul, Artin-Schelter regular of global dimension 4, has the same Hilbert series as the polynomial ring on 4 variables, satisfies the $\chi$ condition, and so on. These results are special cases of general results proved for a triple $(A,T,H)$ consisting of a Hopf algebra $H$, an (often graded) $H$-comodule algebra $A$, and an $H$-torsor $T$. Those general results involve transferring properties between $A$, $A \otimes T$, and $(A \otimes T)^\textrm {{co} H}$. We then investigate $\widetilde {A}$ from the point of view of non-commutative projective geometry. We examine its point modules, line modules, and a certain quotient $\widetilde {B}:=\widetilde {A}/(\Theta ,\Theta ’)$ where $\Theta$ and $\Theta ’$ are homogeneous central elements of degree two. In doing this we show that $\widetilde {A}$ differs from $A$ in interesting ways. For example, the point modules for $A$ are parametrized by $E$ and 4 more points, whereas $\widetilde {A}$ has exactly 20 point modules. Although $\widetilde {B}$ is not a twisted homogeneous coordinate ring in the sense of Artin and Van den Bergh, a certain quotient of the category of graded $\widetilde {B}$-modules is equivalent to the category of quasi-coherent sheaves on the curve $E/E[2]$ where $E[2]$ is the 2-torsion subgroup. We construct line modules for $\widetilde {A}$ that are parametrized by the disjoint union $(E/\langle \xi _1\rangle ) \sqcup (E/\langle \xi _2\rangle ) \sqcup (E/\langle \xi _3\rangle )$ of the quotients of $E$ by its three subgroups of order 2.

Isomorphisms of Nonnoetherian Down-Up Algebras

Fil: Chouhy, Sergio Nicolas. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Ciudad Universitaria. Instituto de Investigaciones Matematicas Luis A. Santalo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matematicas Luis A. Santalo; Argentina

Finite dimensional Hopf actions on algebraic quantizations

Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra [arxiv.org/abs/1409.1644, arxiv.org/abs/1509.01165], we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra k[z_1,...,z_s] factors through a group action, and this in fact holds for any finite dimensional Hopf action if s=0. We also generalized these results to finite dimensional Hopf actions on algebras of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations of commutative domains. This includes universal enveloping algebras of finite dimensional Lie algebras, spherical symplectic reflection algebras, quantum Hamiltonian reductions of Weyl algebras (in particular, quantized quiver varieties), finite W-algebras and their central reductions, quantum polynomial algebras, twisted homogeneous coordinate rings of abelian varieties, and Sklyanin algebras. The generalization in the last three cases uses a result from algebraic number theory, due to A. Perucca.

The point variety of quantum polynomial rings

We show that the reduced point variety of a quantum polynomial algebra is the union of specific linear subspaces in Pn; we describe its irreducible components and give a combinatorial description of the possible configurations in small dimensions.

Hochschild cohomology and quantum Drinfeld Hecke algebras

Quantum Drinfeld Hecke algebras are generalizations of Drinfeld Hecke algebras in which polynomial rings are replaced by quantum polynomial rings. We identify these algebras as deformations of skew group algebras, giving an explicit connection to Hochschild cohomology. We compute the relevant part of Hochschild cohomology for actions of many reflection groups, and we exploit computations from Naidu et al. (Proc Am Math Soc 139:1553–1567, 2011) for diagonal actions. By combining our work with recent results of Levandovskyy and Shepler (Can J Math 66:874–901, 2014) we produce examples of quantum Drinfeld Hecke algebras. These algebras generalize the braided Cherednik algebras of Bazlov and Berenstein (Selecta Math 14(3–4):325–372, 2009).

The Feigin Tetrahedron

The first goal of this note is to extend the well-known Feigin homomorphisms taking quantum groups to quantum polynomial algebras. More precisely, we define generalized Feigin homomorphisms from a quantum shuffle algebra to quantum polynomial algebras which extend the classical Feigin homomorphisms along the embedding of the quantum group into said quantum shuffle algebra. In a recent work of Berenstein and the author, analogous extensions of Feigin homomorphisms from the dual Hall-Ringel algebra of a valued quiver to quantum polynomial algebras were defined. To relate these constructions, we establish a homomorphism, dubbed the quantum shuffle character, from the dual Hall-Ringel algebra to the quantum shuffle algebra which relates the generalized Feigin homomorphisms. These constructions can be compactly described by a commuting tetrahedron of maps beginning with the quantum group and terminating in a quantum polynomial algebra. The second goal in this project is to better understand the dual canonical basis conjecture for skew-symmetrizable quantum cluster algebras. In the symmetrizable types it is known that dual canonical basis elements need not have positive multiplicative structure constants, while this is still suspected to hold for skew-symmetrizable quantum cluster algebras. We propose an alternate conjecture for the symmetrizable types: the cluster monomials should correspond to irreducible characters of a KLR algebra. Indeed, the main conjecture of this note would establish this "KLR conjecture" for acyclic skew-symmetrizable quantum cluster algebras: that is, we conjecture that the images of rigid representations under the quantum shuffle character give irreducible characters for KLR algebras.

Automorphisms for Skew PBW Extensions and Skew Quantum Polynomial Rings

In this work, I study the automorphisms of skew PBW extensions and skew quantum polynomials. I use Artamonov's works as reference for getting the principal results about automorphisms for generic skew PBW extensions and generic skew quantum polynomials. In general, if I have an endomorphism on a generic skew PBW extension and there are some x i , x j , x u such that the endomorphism is not zero on these elements and the principal coefficients are invertible, then endomorphisms act over x i as a i x i for some a i in the ring of coefficients. Of course, this is valid for quantum polynomial rings, with r = 0, as such Artamonov shows in his work. We use this result for giving some more general results for skew PBW extensions, using filtred-graded techniques. Finally, I use localization to characterize some class the endomorphisms and automorphisms for skew PBW extensions and skew quantum polynomials over Ore domains.

Quantum Drinfeld Orbifold Algebras

Quantum Drinfeld orbifold algebras are the generalizations of Drinfeld orbifold algebras, which are obtained by replacing polynomial rings by quantum polynomial rings. In [6], the authors give necessary and sufficient conditions on a defining parameters to obtain Drinfeld orbifold algebras. In this article we generalize their result. It also simultaneously generalizes the result of [5] about quantum Drinfeld Hecke algebras.

Quantum Drinfeld Hecke Algebras

AbstractWe consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a Poincaré–Birkhoff–Witt property using the theory of noncommutative Gröbner bases. We include applications to the case of abelian groups and the case of groups acting on coordinate rings of quantum planes. In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology gives an elegant description of the Poincaré–Birkhoff–Witt conditions.

Integral calculus on quantum exterior algebras

Hom-connections and associated integral forms have been introduced and studied by Brzeziński as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on a differential calculus (Ω, d) over an algebra A yields the integral complex which for various algebras has been shown to be isomorphic to the non-commutative de Rham complex (in the sense of Brzeziński et al. [Non-commutative integral forms and twisted multi-derivations, J. Noncommut. Geom.4 (2010) 281–312]). In this paper we shed further light on the question when the integral and the de Rham complex are isomorphic for an algebra A with a flat Hom-connection. We specialize our study to the case where an n-dimensional differential calculus can be constructed on a quantum exterior algebra over an A-bimodule. Criteria are given for free bimodules with diagonal or upper-triangular bimodule structure. Our results are illustrated for a differential calculus on a multivariate quantum polynomial algebra and for a differential calculus on Manin's quantum n-space.

B-Construction and C-Construction

B-construction is a way of obtaining a graded algebra from the triple consisting of an additive category, an object, and an autoequivalence, while C-construction is a way of obtaining an algebra (without unity) from the pair consisting of an additive category and a set of objects. In this article, we study and compare three important classes of algebras in noncommutative algebraic geometry and representation theory of finite dimensional algebras, namely, quantum polynomial algebras, preprojetive algebras and trivial extensions, via these constructions.

Quantum Sphere $${\mathbb{S}^4}$$ as a Non-Levi Conjugacy Class

We construct a U_h(sp(4))-equivariant quantization of the four-dimensional complex sphere S^4 regarded as a conjugacy class, Sp(4)/Sp(2)x Sp(2), of a simple complex group with non-Levi isotropy subgroup, through an operator realization of the quantum polynomial algebra C_h[S^4] on a highest weight module of U_h(sp(4)).

Twisted rings and moduli stacks of “fat” point modules in non-commutative projective geometry

The Hilbert scheme of point modules was introduced by Artin–Tate–Van den Bergh to study non-commutative graded algebras. The key tool is the construction of a map from the algebra to a twisted ring on this Hilbert scheme. In this paper, we study moduli stacks of more general “fat” point modules, and show that there is a similar map to a twisted ring associated to the stack. This is used to provide a sufficient criterion for a non-commutative projective surface to be birationally PI. It is hoped that such a criterion will be useful in understanding Mike Artinʼs conjecture on the birational classification of non-commutative surfaces.

The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings

Given a projective scheme X over a field k, an automorphism σ: X → X, and a σ-ample invertible sheaf L, one may form the twisted homogeneous coordinate ring B = B(X, L, σ), one of the most fundamental constructions in noncommutative projective algebraic geometry. We study the primitive spectrum of B, as well as that of other closely related algebras such as skew and skew-Laurent extensions of commutative algebras. Over an algebraically closed, uncountable field k of characteristic zero, we prove that the primitive ideals of B are characterized by the usual Dixmier-Moeglin conditions whenever dim X ≤ 2.

Asymmetry of Ext-groups

In this paper, we will use techniques of noncommutative projective geometry to construct examples of algebras R over a field k not satisfying the following two types of symmetric behaviors of Ext-groups: ( EE) For any pair of finitely generated R-modules ( M , N ) , dim k Ext R i ( M , N ) < ∞ for all i ∈ N if and only if dim k Ext R i ( N , M ) < ∞ for all i ∈ N . ( ee) For any pair of finitely generated R-modules ( M , N ) , Ext R i ( M , N ) = 0 for all i ≫ 0 if and only if Ext R i ( N , M ) = 0 for all i ≫ 0 . In particular, and contrary to the commutative case, we give a simple example of a noncommutative noetherian Gorenstein (Frobenius) local algebra satisfying ( uac) (uniform Auslander condition) but not satisfying ( ee).