Schedule for: 24w5505 - Poisson Geometry and Artin-Schelter Regular Algebras

A set dinner is served daily between 5:30pm and 7:30pm in the Xianghu Lake National Tourist Resort.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

For any positively filtered algebra, the property of skew Calabi-Yau or having Van den Bergh duality can be lifted as usual, but not for Calabi-Yau property. Calabi-Yau property often emerges form the deformation of unimodular Poisson structure. Suppose A is a filtered algebra such that the associated graded algebra gr(A) is commutative Calabi-Yau. Then gr(A) has a canonical Poisson structure with a modular derivation. We describe the connection between the Nakayama automorphism of A and the modular derivation of gr(A) by using homological determinants as a bridge. In particular, it is proved that A is Calabi-Yau if and only if gr(A) is unimodular as Poisson algebra under some mild assumptions. As an application, we derive that the ring of differential operators over a smooth variety is Calabi-Yau. I will start from the definitions of (skew) Calabi-Yau algebras and homological determinants of (Hopf) actions on them. Some applications will also be given in the talk. This talk is based on a joint work with Ruipeng Zhu.

We present a general framework for constructing polynomial integrable systems with respect to linearizations of Poisson varieties that admit log-canonical coordinate systems. Our construction is in particular applicable to Poisson varieties with compatible cluster or generalized cluster structures. As special cases, we consider an arbitrary standard complex semi- simple Poisson Lie group G or Schubert cells in the flag variety of G, equipped with the standard Poisson structures and the compatible Berenstein-Fomin-Zelevinsky cluster structures, as well as the dual Poisson Lie group of GL(n, C) equipped with the Gekhtman-Shapiro-Vainshtein generalized cluster structure. In each of the three cases, we show that every extended cluster in the respective cluster structure gives rise to a polynomial integrable system with respect to linearizations of the Poisson structures. This is joint work with Yanpeng Li and Yu Li.

Lunch is served daily between 11:30am and 1:30pm in the Xianghu Lake National Tourist Resort

I will discuss how to construct a large collection of “quantum projective spaces”, in the form of Koszul, Artin-Schelter regular quadratic algebras with the Hilbert series of a polynomial ring. I will do so by starting with the toric ones (the q-polynomial algebras), and then deforming their relations using a diagrammatic calculus, proving unobstructedness of such deformations under suitable nondegeneracy conditions. Time permitting, I will show that these algebras coincide with the canonical quantizations of corresponding families of quadratic Poisson structures. This provides new evidence to Kontsevich's conjecture about convergence of his deformation quantization formula. This is joint work with Brent Pym and Travis Schedler.

This talk is based on joint work with Alex Chirvasitu and S. Paul Smith. Feigin and Odesskii's elliptic algebras constitute a deformation of the polynomial algebra, which induces a Poisson structure on projective space. Hua and Polishchuk showed that this Poisson structure is the same as the one Polishchuk defined in abstract terms. We describe the symplectic leaves for this Poisson structure in terms of higher secant varieties to an elliptic curve.

The ozone group of an associative algebra $A$ is defined as the group of automorphisms of $A$ which fix every element of its center. The ozone group has been utilized to study the center of PI skew polynomial rings, and to characterize skew polynomial rings in the class of connected graded algebras. In this talk I will discuss work on adapting these ideas to polynomial Poisson algebras in positive characteristic. This includes connections between the ozone group and the unimodularity condition, centers of skew symmetric Poisson structures, and work on characterizing skew symmetric Poisson structures. This is joint work with Kenneth Chan, Robert Won, and James J. Zhang.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

The integral is an important structure in a finite-dimensional Hopf algebra. Lu, Wu, and Zhang generalized this to define a homological integral for any Artin-Schelter Gorenstein Hopf algebra. This homological integral has many applications in the study of Hopf algebras of small GK-dimension. A weak Hopf algebra is a generalization of a Hopf algebra in which the comultiplication does not necessarily preserve the unit, and the counit is not necessarily multiplicative, but weaker axioms are satisfied. Weak Hopf algebras arise naturally in the study of tensor categories, for example. We report on joint work with Rob Won and James Zhang that shows how to define a homological integral for an AS Gorenstein weak Hopf algebra, and discuss its applications.

This is a report on my current work with Alejandro Cabrera (Rio de Janeiro) on star products given by semi-classical Fourier integral operators. I will sketch our definition of these types of star products and discuss our main result: the Lagrangian underlying such a star product is the graph of the multiplication of a local symplectic groupoid integrating the deformed Poisson structure. As a consequence, I will argue that, in general, one should expect quantization to involve partial algebras rather than algebras.

In his Master's Thesis, Vashaw identified two interesting quadratic AS-regular algebras of dimension four, which we call R and S, which were graded by groups of order 16 such that the identity component was also an AS-regular algebra. In joint work with Goetz, Kirkman, and Vashaw, we study the geometry (namely, the point and line schemes and their incidence relations) of the algebras R and S, as well as an algebra related to S which we denote by T.

Let $H$ be a semisimple Hopf algebra, and let $e$ be the integral of $H$ such that $\varepsilon(e)=1$. Suppose that $H$ acts on an algebra $A$. Let $A\#H$ be the smash product, and let $A^H$ be the invariant subalgebra of $A$. There is a natural $A\#H$-bimodule map $\beta_{A,H}:A\otimes_{A^H} A\longrightarrow A\#H$, defined by $\beta_{A,H}(a\otimes b)=(a\#e)(1\#b)$. We call $\beta_{A,H}$ the adjunction map associated to the $H$-action on $A$. In this talk, I will report some categorical properties of $A^H$ determined by $\beta_{A,H}$.

In this talk, I will discuss derivations of a class of noncommutative polynomial algebras, the so-called, quantum nilpotent algebras, and their primitive quotients. This is joint work in progress with Samuel Lopes (Porto) and Isaac Oppong (Greenwich).

In the early 1990s, Artin, Tate, and Van den Bergh conjectured that all Artin-Schelter regular algebras are domains. To date there are not many available tools to tackle this conjecture, even in the special case of Koszul algebras. In this talk I will discuss joint work with Daniel Rogalski that provides one such tool that may be helpful: a necessary and sufficient condition for a Koszul algebra to be a domain. The condition is stated in terms of properties of syzygy modules over the quadratic dual algebra. These techniques are also sufficient to prove that graded twisted CY-2 algebras defined by many quivers are prime.

I will survey some aspects of the theory of holomorphic Poisson brackets on projective algebraic varieties and related noncommutative deformations. In particular, some conjectures about these objects will be discussed.

Noncommutative deformations of the projective plane and those of the quadric are obtained from 3-dimensional AS-regular quadratic Z-algebras and cubic Z-algebras, respectively. In this talk I will explain how these classes of Z-algebras arise from helices of the derived category of the commutative P2 and the commutative quadric, respectively. This observation allows us to introduce (infinitely) many classes of AS-regular Z-algebras, with which we can cover all deformation types of del Pezzo surfaces equally well. I will illustrate this with some examples and then try to summarize what we know about and expect from these Z-algebras.

This talk is based on a joint work with Adam Nyman. In this talk, we will define an ASF-regular Z-algebra and characterize a noncommutative projective scheme associated to an ASF-regular Z-algebra. As an application, we will show that a skew quadric hypersurface has an ASF-regular Z-algebra as a homogeneous coordinate ring. If time permits, we will discuss noncommutative conics and noncommutative quadric surfaces.

We will discuss Poisson valuation and its application in computing Poisson automorphism groups and other related topics. It is a joint work with Hongdi Huang, Xin Tang, and James Zhang.

Connected Hopf algebras of finite Gelfand-Kirillov dimension (over a field of characteristic zero) can be viewed as generalizations of the universal enveloping algebras of finite dimensional Lie algebras as well as the noncommutative counterpart of unipotent algebraic groups. They enjoy many nice ring-theoretical and homological properties. Particularly, they are Noetherian domains, Artin-Schelter regular and twisted Calabi-Yau. In this talk, I will discuss the structure of them and their coideal subalgebras from the perspective of Ore extensions.

We will present results on the computation of Poisson cohomology groups for several classes of unimodular Poisson polynomial algebras in three variables. If time permits, we will also discuss a couple of technical tools introduced to aid the computation. This is joing work with Hongdi Huang, Xingting Wang and James Zhang.

The ring of differential operators on a cuspidal curve whose coordinate ring is a numerical semigroup algebra is shown to be a cocommutative and cocomplete left Hopf algebroid, which essentially means that the category of D-modules is closed monoidal. If the semigroup is symmetric so that the curve is Gorenstein, it is a full Hopf algebroid (admits an antipode), which means that the subcategory of those D-modules that are finite rank vector bundles over the curve is rigid. Based on joint work with Myriam Mahaman.

Moduli spaces of stable sheaves on Fano threefolds are known to exhibit pathological behavior in general. Meanwhile, for certain specific cases—such as ideal sheaves of curves with small degree and genus in the cubic threefold, or moduli spaces of lower-rank ACM bundles—these spaces are well-behaved. From a modern derived categorical perspective, we have the so-called Kuznetsov component $Ku(X)$ in $D(X)$. The well-behaved moduli spaces mentioned above actually parametrize stable objects within $Ku(X)$. In this talk, I will begin by recapping this framework with a detailed overview of known results. I will then present our recent work on higher-dimensional moduli spaces of stable objects in $Ku(X)$.

Results of Etingof and Walton show that there are algebras $A$ (e.g. commutative domains) with no quantum symmetries, i.e. if $H$ is a semisimple Hopf algebra acting inner faithfully on $A$, then $H$ is a group algebra. We discuss circumstances, where given a Hopf algebra $H$, an AS regular algebra $A$ that supports a non-trivial $H$ action can be constructed. In some cases the subalgebra of invariants is also AS regular.

We present a construction of weak graded Lie 2-algebras associated with quasi-Poisson groupoids. We also establish a morphism between this weak graded Lie 2-algebra of multiplicative forms and the strict graded Lie 2-algebra of multiplicative multivector fields, allowing us to compare and relate different aspects of Lie 2-algebra theory within the context of quasi-Poisson geometry. As an infinitesimal analogy, we explicitly determine the associated weak graded Lie 2-algebra structure of IM forms for any quasi-Lie bialgebroid. This is joint work with Zhuo Chen and Zhangju Liu.