Schedule for: 16w5088 - Bridges between Noncommutative Algebra and Algebraic Geometry

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will discuss recent joint work with Alexandru Chirvasitu and Xingting Wang (arXiv:1605.06428), where we define the universal quantum group $\mathcal{H}$ that preserves a pair of Hopf comodule maps whose underlying vector space maps are preregular forms defined on dual vector spaces. This generalizes the construction of Bichon and Dubois-Violette (2013), where the target of these comodule maps are the ground field. We also recover the quantum groups introduced by Dubois-Violette and Launer (1990), by Takeuchi (1990), by Artin, Schelter, and Tate (1991), and by Mrozinski (2014), via our construction. As a consequence, we obtain an explicit presentation of a universal quantum group that coacts simultaneously on a pair of N-Koszul Artin-Schelter regular algebras with arbitrary quantum determinant.

Support varieties are a tool providing some information about representations. They were first introduced by Quillen for finite groups, and have since been generalized in several directions, including to finite dimensional self-injective algebras (under some finiteness assumptions). One asks which properties of support varieties for finite group representations are true more generally? In this talk, we will give an overview of the theory of varieties for modules of self-injective algebras. We will focus on the tensor product property for Hopf algebras, that is, that the variety of a tensor product of modules is the intersection of the varieties. We will give examples of Hopf algebras constructed from finite groups for which the tensor product property does not hold.

Let $S$ be the antipode of a Hopf algebra $H$. The gauge invariance of Frobenius-Schur indicators implies that the traces of $S$ and $S^2$ are invariants of the finite tensor category Rep($H$). However, the question of whether the traces of all the powers of $S$ are gauge invariants is generally open. An affirmative answer of this question immediately implies the invariance of the order of $S$. In this talk, we will show that they are invariants of Rep($H$) when the Jacobson radical of $H$ is a Hopf ideal. This is a joint work with Cris Negron.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We introduce a method named homogeneous PBW deformation that preserves the regularity and some other homological properties for multigraded algebras. The method is used to produce Artin–Schelter regular algebras without the hypothesis on grading.

The 3-dimensional Sklyanin algebras degenerate to algebras isomorpic to either $\frac{\mathbb{C} < x,y,z > }{(x^2,y^2,z^2)}$ or $\frac{\mathbb{C} < x,y,z > }{ (xy,yz,zx)}$ due to a result of S. Paul Smith. The Heisenberg group of order 27, $H_3$ acts on these algebras as gradation preserving automorphisms. I will show that, using the representation theory of $H_3$, there exists a 1-dimensional family of quotients $A_t$, $t \in \mathbb{C}$ of these algebras such that $A_t \cong S(V)$ as graded $H_3$-module, where $S(V)$ is the polynomial ring in 3 variables with $V$ the Schrödinger representation of $H_3$. In addition, these quotients are noetherian and have a central element of degree 3, as in the regular case.

The Hochschild cohomology of an abelian category describes the infinitesimal deformation theory of this abelian category in the sense of Lowen--Van den Bergh. Applying this to the category coh X for a variety X gives information on how the noncommutative deformations of this variety behave. In this talk I will explain how to compute the Hochschild cohomology of the abelian category qgr A for a quadratic (resp. cubic) 3-dimensional Artin--Schelter regular algebra, i.e. for a noncommutative projective plane (resp. noncommutative quadric surface). The main ingredients for this are the full and strong exceptional collection in the derived category and the classification of these algebras using the geometry of the point scheme.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Let ${\Bbbk}$ be an algebraically closed field of characteristic zero. Maurice Auslander proved that when a finite subgroup $G$ of GL$_n(\Bbbk)$, containing no reflections, acts on $A=\Bbbk[x_1,\dots,x_n]$ naturally, with fixed subring $A^G$, then the skew group algebra $A \# G$ is isomorphic to End$_{A^G}(A)$ as algebras. There are recent results extending Auslander's Theorem to non(co)commutative settings of actions on Artin-Schelter regular algebras $A$ by groups or Hopf algebras that contain no ``reflections''. Bao, He, and Zhang develop the notion of pertinency, and apply it to prove Auslander's Theorem for certain group actions on $\Bbbk_{-1}[x_1, \dots, x_n]$, on $U(\mathfrak{g})$ for $\mathfrak{g}$ finite dimensional, and on certain classes of noetherian down-up algebras. Work in progress with Gaddis and Moore proves Auslander's Theorem for the permutation action of $S_n$ on $\Bbbk_{-1}[x_1, \dots, x_n]$ for n= 3 and 4. Work with Chan, Walton and Zhang proves Auslander's theorem when $A$ is an AS regular algebra of dimension 2 and $H$ is a semisimple Hopf algebra acting on $A$ so that $A$ is a graded $H$-module algebra under an action that is inner faithful and has trivial homological determinant. With Chen and Zhang we prove Auslander's Theorem for homogeneous, inner-faithful group coactions on noetherian down-up algebras with trivial homological determinant.

We survey some recent progress on the Hochschild cohomology of global quotient orbifolds, with a focus on works of Travis Schedler, Sarah Witherspoon and myself. A global quotient orbifold is, for us, the stack quotient of a quasi-projective scheme by a finite group in characteristic $0$. The vector space structure on the cohomology of such an object was given in a recent paper of Arinkin, C\u{a}ld\u{a}raru, and Hablicsek, and, in the case in which the scheme is affine space, a complete description of the Gerstenhaber structure was given by Shepler, Witherspoon, and myself. One finds that the Hochschild cohomology and its algebraic structures can be described in terms of the geometries of the fixed spaces under the actions of individual group elements.

Similar to the modular vector fields in Poisson geometry, modular derivations can be defined for smooth Poisson algebras with trivial canonical bundle. By twisting Poisson modules with the modular derivation, the Poisson cochain complex with values in any Poisson module is isomorphic to the Poisson chain complex with values in the corresponding twisted Poisson module. Then a version of twisted Poincar\'{e} duality is deduced between Poisson homologies and Poisson cohomologies. If the Poisson structure is pseudo-unimodular, then its Poisson cohomology as Gerstenhaber algebra is exact, that is, it has a Batalin-Vilkovisky algebra structure by using the isomorphism between the Poisson cochain complex and chain complex.

In 1981, Laffey described subalgebras of associative algebras generated by two idempotents, and showed that central simple algebras (other than division algebras) can be generated by three idempotents. We describe his structure more precisely, as a direct product of Azumaya algebras and a PI-algebra satisfying the identity $(xy - yx)^n,$ and, following an idea of Shestakov, show that Jordan subalgebras of a finite dimensional algebra generated by $n$ completely primitive idempotents algebraic are of dimension $\le 2^n -1.$

Discriminants play a key role in the study of PI algebras (PIAs): orders in central simple algebras, Azumaya loci of PIAs, the isomorphism and automorphism problems for PIAs. Previously, they were computed for very few PI algebras. We will present a general method for computing discriminants of PIAs which is applicable to algebras obtained by specialization from families, such as quantum algebras at roots of unity. It relies on a connection with Poisson geometry. From a different perspective the technique builds a bridge to the theory of discriminants of number fields, where factorizations into primes are replaced by factorizations into Poisson primes. This is a joint work with Jesse Levitt, Bach Nguyen and Kurt Trampel.

My talk, representing joint work with Birge Huisgen-Zimmerman, addresses finite dimensional algebras over an algebraically closed field. We describe closures of representation-theoretically crucial locally closed subsets of the parametrizing varieties for the representations with fixed dimension vector. The description leads to a novel module invariant which is upper semicontinuous on the parametrizing varieties. This invariant, in turn, is instrumental in completing the classification of the irreducible components over arbitrary truncated path algebras.

Van den Bergh defined a notion of blowing up a noncommutative surface at a point lying on a commutative divisor in the surface. The definition is made by constructing a Rees ring in a certain category of functors with adjoints. We show how to make the adjoint functors more explicit using what is known as projective effacements. This allows us to define blowups more generally, and also helps to make it clear that the coordinate rings of Van den Bergh's blowups of the Sklyanin projective plane are the same as the ring-theoretic blowups studied by Sierra, Stafford, and the author.

Let $k$ be an algebraically closed field of characteristic zero. For any positive integer $n$, we construct a Calabi-Yau algebra $R(n)$, which induces a Poisson deformation of $k[x_1, \dots, x_n]$ and generalises a construction given by Pym when $n=4$. Modulo scalars, the graded automorphism group of $R(n)$ is isomorphic to $k$, and we consider not only $R(n)$ but its Zhang twist $R(a,n)$ by the automorphism corresponding to $a$. Each $R(a,n)$ induces a Poisson structure on $k[x_1, \dots, x_n]$ in the semiclassical limit, and we study this structure. We show that the Poisson spectrum of the limit is homeomorphic to $\operatorname{Spec} R(a,n)$, and explicitly describe $\operatorname{Spec} R(a,n)$ as a union of commutative strata. This is joint work with Cesar Lecoutre.

This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls. Let $G$ be a finite subgroup of $GL(n,K)$ for a field $K$ whose characteristic does not divide the order of $G$. The group $G$ acts linearly on the polynomial ring $S$ in $n$ variables over $K$. When $G$ is generated by reflections, then the discriminant $D$ of the group action of $G$ on $S$ is a hypersurface with a singular locus of codimension 1. In this talk we give a natural construction of a noncommutative resolution of singularities of the coordinate ring of $D$ as a quotient of the skew group ring $A=G*S$. We will explain this construction, which gives a new view on Knörrer's periodicity theorem for matrix factorizations and allows to extend Auslander's theorem about the algebraic version of the McKay correspondence to reflection groups.

I will discuss various results/open questions on the prime ideal structure of the quantum grassmannian and their links with the totally nonnegative grassmannian.

If $D/F$ is a division algebra of degree 3, then the Severi-Brauer variety of $D$, call it $X$, is a form of the projective plane. The line bundle $O(3)$ is defined on $X$, which says it makes sense to talk about cubic curves on $X$. Since $X$ has no rational points, these are genus one curve and not elliptic curves. However, they are principle homogeneous spaces over their Jacobians $E$, which are elliptic curves. Which ones occur?

It is known that a connected graded algebra is Artin-Schelter (AS) regular if and only if it is twisted Calabi-Yau (CY). While AS regular algebras are necessarily connected, a twisted CY algebra need not be. Thus we ask: for algebras that are graded but not necessarily connected, is the twisted CY property equivalent to a suitable analogue of the AS regular property? We give a positive answer, using generalized AS regular properties inspired by the work of Martinez-Villa, Minamoto, and Mori. We will also describe the structure and properties of graded twisted CY algebras in very low dimension (at most 2). This is a preliminary report on joint work with Daniel Rogalski.

Suppose $\phi$ is a birational automorphism of a projective variety $X$ over $\overline{Q}$. If $\phi$ preserves a rational fibration $X \to Y$ and $\dim Y > 0$, then it is impossible for there to be a $\overline{Q}$-point of $X$ with Zariski dense orbit. It has been separately conjectured by Zhang, Amerik, and Medvedev-Scanlon that this is the only obstruction to the existence of a $\overline{Q}$-point with Zariski dense orbit. We prove their conjecture in positive Kodaira dimension and then, contingent on conjectures in the Minimal Model Program, prove the conjecture for threefolds of Kodaira dimension 0. This is joint work with Jason Bell, Dragos Ghioca, and Zinovy Reichstein.

Let A be a commutative ring, and let \a := \frak{a} be a finitely generated ideal in it. It is known that a necessary and sufficient condition for the derived \a-torsion and the derived \a-adic completion functors to be nicely behaved is the weak proregularity of the ideal \a. In particular, the MGM Equivalence holds under this condition. Because weak proregularity is defined in terms of elements of the ring (specifically, it involves limits Koszul complexes), it is not suitable for noncommutative ring theory. Consider a torsion class T in the category M(A) of left modules over a ring A. We introduce a new condition on T: weak stability. Our first main theorem is that when A is commutative, an ideal \a in A is weakly proregular if and only if the corresponding torsion class T in M(A) is weakly stable. It turns out that when the ring A is noncommutative, one must impose two more conditions on the torsion class T: quasi-compactness and finite dimensionality (these are new names for old conditions). We prove that for a torsion class T that is weakly stable, quasi-compact and finite dimensional, the right derived T-torsion functor is isomorphic to a left derived tensor functor. This result involves derived categories of bimodules. Some examples will be given. The third main theorem is the Noncommutative MGM Equivalence, under the same assumptions on T. Finally, there is a theorem about derived left-sided and right-sided torsion for complexes of bimodules. This last theorem is a generalization of a result of Van den Bergh from 1997, and it corrects an error in a paper of Yekutieli-Zhang from 2003. We expect that the approach outlined in this talk will open up the way to a useful theory of rigid dualizing complexes in the arithmetic noncommutative setting (namely without a base field). The work above is joint with Rishi Vyas.

AS-regular algebras are non-commutative analogues of smooth projective schemes, with those of global dimension four behaving in many ways like three-dimensional projective space. In this talk I will introduce a specific family of such algebras arising from certain elliptic solutions of the quantum Yang-Baxter equation and study the phenomenon whereby a quantum group acts on each algebra in the family. The quantum group action gives rise to autoequivalences of the category of (graded) modules that do not come from genuine algebra automorphisms. This then helps in classifying certain well-behaved modules that play the role of lines inside the quantum projective space. (joint w/ S. Paul Smith)

Let G be a compact p-adic analytic group, k a finite field of characteristic p, and kG the completed group algebra. We outline some of what is known about the structure of this ring and its minimal prime ideals, and sketch a proof that, in the case when G is virtually nilpotent, kG is a catenary ring.

This talk is based on a joint work with S. P. Smith. AS-regular algebras is an important class of algebras to study in noncommutative algebraic geometry. If S is an m-Koszul AS-regular algebra, then it was observed by several people that S is determined by a twisted superpotential. In this talk, we will see that such a twisted superpotential is uniquely determined by S up to non-zero scalar multiples and plays a crucial role in studying S. In particular, we will see in this talk that, using the twisted superpotential w_S associated to S, we can compute (1) the Nakayama automorphism of S, (2) a graded algebra automorphism of S, and (3) the homological determinant of a graded algebra automorphism of S (which is an essential ingredient for invariant theory of an AS-regular algebra). If time permits, we will present some applications to 3-dimensional noetherian Calabi-Yau algebras (which is partially based on a joint work with K. Ueyama).

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.