Schedule for: 22w5084 - Noncommutative Geometry and Noncommutative Invariant Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We review some open questions, conjectures, and important on-going projects in noncommutative algebra.

Let $A$ be a noetherian connected graded $\Bbbk$-algebra with a balanced dualizing complex, and let $X$ be a cochain complex of graded left $A$-modules. The elements of $X$ possess both an internal and various homological degrees, and it is useful to study the relationships between these degrees. Jörgensen and Dong-Wu extended the study of Tor-regularity and Castelnuovo-Mumford regularity from commutative algebras to noncommutative algebras. We consider these regularities further, and define new numerical invariants that involve linear combinations of internal and homological degrees. This is joint work with Robert Won and James J. Zhang.

It has been observed that Poisson algebras are closely related to their quantizations in many perspectives. In this talk, we will paint a similar picture in terms of the automorphisms and isomorphisms for several classes of Poisson algebras and compare the results with their quantum analogues. Some of the results are ongoing joint work with Xingting Wang and James Zhang.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk 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!

In this talk, we will explore how a twist of an algebra's multiplicative structure by an automorphism, can be extended to a twist of certain Hopf algebras. We do so by twisting a bialgebra and by lifting it to a Hopf algebra using Takeuchi's Hopf envelope construction. Moreover, we examine when our construction coincides with a 2-cocycle twist of the Hopf algebra. We analyze our work in the context of Manin's universal quantum groups and solutions to the quantum Yang Baxter equation. This is joint work with H. Huang, V, C. Nguyen, C. Ure, K. Vashaw, and X. Wang.

I will survey results on the "finite generation conjecture" (FGC) for finite-dimensional Hopf algebras. The FGC proposes that cohomology over such a Hopf algebra H enjoys many global finiteness properties which imply, for example, that all extension algebras Ext*_H(V,V) are finitely generated and finite over their centers. If time permits, I will describe some advanced interpretations of Deligne's conjecture and their relations to tensor triangular geometry.

For any Hopf algebra $H$ and any 2-cocycle $\sigma$ on $H$, the twist of $H^\sigma$ arises by deforming the underlying algebra structure. It is known that $H$ and $H^\sigma$ are Morita-Takeuchi equivalent. In particular, for any $H$-comodule algebra $A$, there is a twisted $H^\sigma$-comodule algebra $A_{\sigma^{-1}}$. In this talk, I will explain how this twisting may be thought of as an extension of twisting $A$ by a graded automorphism. As an example, I will consider twisting of preregular forms and their associated superpotential algebras by 2-cocycles. This is joint work with Hongdi Huang, Van Nguyen, Kent Vashaw, Padmini Veerapen, and Xingting Wang.

In this talk I will discuss Hopf actions in the setting of $\mathbb{Z}$-graded algebras. The Weyl algebra is an example of such an algebra, but has no finite dimensional quantum symmetry. Instead, we study quantum generalized Weyl algebras (GWAs), which exhibit actions by generalized Taft algebras that respect their $\mathbb{Z}$-grading. These actions are extensions, or `quantum thickenings', of cyclic group actions. This is joint work with Robert Won.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

Root of unity quantum cluster algebras form a vast class of algebras containing many important subclasses of quantum algebras at roots of unity arising in Lie theory and topology. Using Cayley-Hamilton algebras in the sense of Procesi, one shows that they contain canonical central subalgebras, isomorphic to the underlying classical cluster algebras with the property that the root of unity algebra is module finite over the central subalgebra. We will present results that explicitly describe the fully Azumaya loci of each root of unity quantum cluster algebra. We will also show that the spectrum of the underlying cluster algebra has an explicit torus orbit of symplectic leaves with respect to the Gekhtman-Shapiro-Vainshtein Poisson structure. This is a joint work with Greg Muller, Bach Nguyen and Kurt Trampel.

We will discuss the Quantum Riemannian Geometry of the fuzzy sphere, where the fuzzy sphere is defined as the angular momentum algebra $[x_i,x_j]=2\imath\lambda_p \epsilon_{ijk}x_k$ modulo setting $\sum_i x_i^2$ to a constant, using a recently introduced 3D rotationally invariant differential structure. It is found that the metrics are given by symmetric $3 \times 3$ matrices $g$ and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients. As an application, we will discuss the construction of the Euclidean quantum gravity on the fuzzy unit sphere; and also the charge 1 monopole for the 3D differential structure.

In noncommutative projective algebraic geometry, twistings of homogenous coordinate rings give equivalences between noncommutative projective schemes. We introduce a Poisson version of such twisting of any graded Poisson algebra. We show that every graded Poisson algebra is the graded twist of a unimodular one. We also discuss various new concepts in Poisson twisting related to the computation of Poisson homology and Poisson cohomology. This is joint work with Hongdi Huang, Xin Tang and James Zhang.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will discuss the work on the structure of graded unimodular Poisson algebras in dimension 3 when the weights of the three variables are arbitrary. To investigate the related homological properties of these Poisson algebras, we first give the complete classification of weighted potentials when the Jacobian structure is homogeneous of degree zero. Moreover, we will also discuss the classification of the potentials of certain degree that have isolated singularities. This is an ongoing joint work with Xin Tang, Xingting Wang and James Zhang.

Cogroupoids have been used in recent years by Bichon and others as a convenient framework to explore Hopf-Galois objects and Morita-Takeuchi equivalences. In this talk, which will build on the previous talk of Charlotte Ure, we will construct a cogroupoid corresponding to m-linear preregular forms, for all m greater than 1. Using this, we recover concretely a partial result of Radschaelders-Van den Bergh, which gives a Morita-Takeuchi equivalence between universal quantum groups of Artin-Schelter regular algebras of dimension 2. We also show that after settinga quantum determinant equal to 1, we can compute a formula for cocycle twists of a universal quantum group in terms of the twists of a preregular form. This is joint work with Hongdi Huang, Van Nguyen, Charlotte Ure, Padmini Veerapen, and Xingting Wang.

The minimal model program, initially introduced to provide a framework for classifying higher dimensional varieties, has also proved useful for studying noncommutative schemes arising as orders on varieties. In this talk, we will look at recent work on orders on arithmetic surfaces. When the order has prime index p>5, many results from classical surface theory can be recovered such as the existence of terminal resolutions, classification of terminal singularities and Castelnuovo's contraction theorem. However, new phenomena appear which do not occur in the case of surfaces over an algebraically closed field. For example, Castenuovo contractions can now introduce singularities on the centre.

In this talk, we will discuss some recent projects (joint work with Georgia Benkart, Rekha Biswal, Ellen Kirkman, and Jieru Zhu) and open problems in tensor representations of finite-dimensional Hopf algebras

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

We call an algebraic structure H "Hopf-like" if its category of (co)representations is monoidal. If additionally H is cocommutative, examples include cocommutative (weak) Hopf algebras, group/groupoid algebras, and universal enveloping algebras of Lie algebras/(some) Lie algebroids. In this talk I will present (classical and new) results showing that an algebra A is an H-module algebra precisely when there exists a structure preserving map from H to a certain collection of linear endomorphisms of A that has the same structure as H. This yields an equivalence between categorical and representation-theoretic notions of an algebra A admitting an action of H. This is joint work with Hongdi Huang, Elizabeth Wicks and Robert Won.

There are serious obstructions to extending the Zariski spectrum Spec as a functor from commutative rings to noncommutative rings. In an attempt to escape these limitations, we are forced to search for a category of ``noncommutative sets'' that is strictly larger than the classical category of sets. Restricting to cases where the maximal spectrum Max is functorial for commutative algebras, we argue that coalgebras serve as a reasonable approximation to generalized sets, and that the finite dual coalgebra is a suitable quantization of Max. We will discuss how the finite dual behaves under twisted tensor products and how it can be understood relative to the center of an affine noetherian PI algebra. We will close by discussing a conjectural path to quantizing the functor Spec itself.

This talk will explain how Arnold’s results for commutative singularitiescan be extended into the noncommutative setting, with the main result being a classification of certain Jacobi algebras arising from (complete) free algebras. This class includes finite dimensional Jacobi algebras, and also Jacobi algebras of GK dimension one, suitably interpreted. The surprising thing is that a classification should exist at all, and it is even more surprising that ADE enters. I will spend most of my time explaining what the algebras are, what they classify, and how to intrinsically extract ADE information from them. At the end, I’ll briefly explain why I’m really interested in this problem, the connection with different quivers, and the applications of the above classification to curve counting and birational geometry. This is joint work with Gavin Brown.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

This talk bridges between noncommutative geometry and higher category theory. A famous link between the two subjects is given by the DG nerve, which turns a DG category into a quasi-category. In this talk, we will enrich this construction keeping track of the linear features of the DG category. More generally, this leads to a notion of quasi-categories in a monoidal category V, which should model weak enrichment in the category of simplicial V objects. (Joint with Arne Mertens)

(Joint work with Ellen Kirkman and Tolulope Oke) The quantum double $D(H)$ of a Hopf algebra $H$ was originally introduced by Drinfel'd in his study of solutions to the quantum Yang-Baxter equation. We use Witherspoon's calculation of the representation ring of the quantum double $D(G)$ of a finite group $G$ to determine families of inner-faithful representations of the quantum double of some generalized quaternion groups. We examine several such representations in detail, and use them to identify some families of quadratic AS-regular algebras (in fact, double Ore extensions) on which $D(G)$ acts.

I will present examples of smooth noncommutative projective schemes using two classes of algebras, namely skew quadric hypersurfaces and twisted Segre products of Artin-Schelter regular algebras.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We study PI skew polynomial rings and the relationship between their centers, ozone groups, parameters, and some newly defined invariants. We investigate in detail several properties of such algebras and their centers in low dimension. This is joint work with Kenneth Chan, Jason Gaddis, and James J. Zhang.

Universal enveloping algebras of finite-dimensional Lie algebras are fundamental examples of well-behaved noncommutative rings. On the other hand, enveloping algebras of infinite-dimensional Lie algebras remain mysterious. For example, it is widely believed that they are never noetherian, but there are very few examples whose noetherianity is known. In this talk, I will introduce a class of infinite-dimensional Lie algebras known as Krichever-Novikov algebras and talk about a recent proof that their enveloping algebras are not noetherian, providing a new family of non-noetherian universal enveloping algebras.

In the theory of finite-dimensional algebras (or equivalently, quivers with admissible relations), "moduli spaces" are a geometric tool for giving structure to infinite families of isomorphism classes of indecomposable representations. The "tame algebras" are those for which such families can always be described with one parameter from the underlying field. More precisely, their moduli spaces are always projective algebraic curves. These moduli spaces have been explicitly described for certain classes of tame algebras over the past 20 years, and so far in every known case they have turned out to be smooth of genus zero; i.e. isomorphic to the projective line P^1. One might conjecture that is true for all tame algebras. This talk will survey the history of this story and recent additional evidence for this conjecture.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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 explain how to describe the birational geometry, including all (partial) crepant resolutions, of quiver varieties and other GIT quotients satisfying mild assumptions (which also includes also some 3D quotient singularities), in terms of varying the stability condition. I will also outline how to extend to moduli spaces with these as local models, such as moduli of 2CY categories (eg Higgs bundles on closed curves, sheaves on K3 surfaces, etc). This is based on joint work with Bellamy and Craw, and also with Kaplan.

The non-commutative algebras $Q_{n,k}(E,\eta)$, introduced by Feigin and ODesskii in the course of generalizing Sklyanin's work, depend on two coprime integers $n>k\ge 1$, an elliptic curve $E$ and a point $\eta\in E$. The degeneration $\eta\to 0$ collapses $Q_{n,1}(E,\eta)$ to the polynomial ring in $n$ variables, and one obtains in this fashion a homogeneous Poisson bracket on that polynomial ring and hence a Poisson structure on the projective space $\mathbb{P}^{n-1}$. The symplectic leaves attached to that structure have received some attention in the literature, including from Feigin and Odesskii themselves and more recently, Hua and Polishchuk. The talk revolves around various results on these symplectic leaves: their concrete description as moduli spaces of sheaf extensions on the elliptic curve $E$, the attendant realization as GIT quotients, resulting good properties (like smoothness) which follow from this without appealing to the symplectic machinery, etc. (joint with Ryo Kanda and S. Paul Smith)

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

An important open question is whether infinite-dimensional noetherian Hopf algebras have finite injective dimension or even must be Artin-Schelter Gorenstein, as conjectured by Brown and Goodearl. The same question can be asked for Weak Hopf algebras. We describe work which proves the conjecture for weak Hopf algebras H which are finitely generated over an affine center. In addition, we will talk about preliminary results which extend the theory of homological integrals to the setting of weak Hopf algebras, which has numerous applications. This is joint research with Rob Won and James Zhang.