## The University of Leeds Algebra Seminar 2017/18   Tuesdays 15:00 Mall 1 (occasionally different time and place)

The seminar will normally take place in Mall 1 at 15:00 on Tuesdays. After the seminar we will have biscuits and tea (and coffee) in the common room.

## Schedule:

 DATE and TIME and PLACE SPEAKER TITLE October 3, 15:15  Mall 1 Raquel Simoes   (University of Lisbon) Reduction for negative Calabi-Yau triangulated categories October 10, 15:15   Mall 1 Nick Manton   (University of Cambridge) Algebra, Geometry and Skyrmions October 17, 15:15  Mall 1 Karin Baur   (Universität Graz) Dimers with boundary, associated algebras and module categories October 24, 15:15  Mall 1 Emilie Dufresne   (University of Nottingham) Separating invariants and local cohomology October 31, 15:30   Mall 1 Xiaoting Zhang   (University of Uppsala) Pyramids and their applications November 7, 15:15  Mall 1 Susan Sierra   (University of Edinburgh) Chain conditions in the enveloping algebra of the Witt algebra November 14, 15:15  Mall 1 Ana Garcia Elsener   (Universität Graz) Triangles in cluster categories and punctured skein relations November 21, 15:15   Mall 1 Joe Karmazyn   (University of Sheffield) Threefold flops, matrix factorisations, and noncommutative algebras December 5, 15:15  Mall 1 Diane Maclagan (University of Warwick) Tropical ideals December 5, 16:30   Mall 1 Idan Eisner (University of Loughborough) Exotic cluster algebras on simple Lie groups January 23, 15:15  Hillary Place SR (G.18) Sira Gratz (University of Glasgow) Noncrossing partitions and thick subcategories January 23, 16:30   Hillary Place SR (G.18) Jordan McMahon (Universität Graz) Higher Frieze Patterns January 30, 15:15  Hillary Place SR (G.18) Philipp Lampe (Durham University) Factoriality and class groups of cluster algebras February 6, 15:15  Hillary Place SR (G.18) Michael Wemyss (University of Glasgow) Intersections in Tits Cones, and Applications February 13, 15:00  Hillary Place SR (G.18) Lutz Hille (Universität Münster) Exceptional and spherical modules for the Auslander algebra of k[T]/T^t February 20, 15:00  Hillary Place SR (G.18) Brent Pym (University of Edinburgh) Holonomic Poisson manifolds and deformations of elliptic algebras February 27, 15:00  Hillary Place SR (G.18) Arnaud Mortier (Dublin City University) Kontsevich invariants in knot theory March 6, 15:00  TBA Alice Rizzardo (University of Liverpool) Cancelled. March 6, 15:00  Hillary Place SR (G.18) Raphael Bennett-Tennenhaus (University of Leeds) Some generalisations of special biserial algebras March 13, 15:00  TBA David Pauksztello (Lancaster University) Cancelled. April 17, 15:00  Mall 1 Milena Hering (University of Edinburgh) Cancelled. April 24, 16:30   Mall 1 Hendrik Suess (University of Manchester) Frobenius splitting of toric and T-varieties May 1, 15:00  Mall 1 Amit Hazi (University of Leeds) $p$-adic expansions of affine type Soergel bimodules May 15, 15:00  Mall 1 David Pauksztello (Lancaster University) Silting theory and stability spaces May 22, 15:00  Mall 1 Alice Rizzardo (University of Liverpool) Enhancements in derived and triangulated categories May 22, 16:15  Mall 1 Colin Ingalls (Carleton University) McKay quivers

## Abstracts

Raquel Simoes (University of Lisbon): Reduction for negative Calabi-Yau triangulated categories Iyama and Yoshino introduced a tool, now known as Iyama-Yoshino reduction, which is very useful in studying the generators and decompositions of positive Calabi-Yau triangulated categories. However, this technique does not preserve the required properties for negative Calabi-Yau triangulated categories. In this talk, we establish a Calabi-Yau reduction theorem for this class of categories. This will be a report on joint work with David Pauksztello.

Nick Manton (University of Cambridge): Algebra, Geometry and Skyrmions Skyrmions arise in a model devised to explain atomic nuclei. Mathematically, they are spatially localised structures that solve certain PDEs, and quantum mechanics also comes in. However, this talk will stress neither the physics nor the PDEs. Instead it will be mainly about the geometrical structure of the Skyrmions. They resemble three-dimensional clusters of particles on the vertices of a Face-Centred-Cubic lattice, and those that are energetically favoured often have the shapes of SU(4) weight diagrams. I will present some (possibly) new results about the weight diagrams, and say something about what this implies for the physics.

Karin Baur (Universität Graz): Dimers with boundary, associated algebras and module categories Dimer models with boundary were introduced in joint work with King and Marsh as a natural generalisation of dimers. We use these to derive certain infinite dimensional algebras and consider idempotent subalgebras w.r.t. the boundary. The dimer models can be embedded in a surface with boundary. In the disk case, the maximal CM modules over the boundary algebra are a Frobenius category which categorifies the cluster structure of the Grassmannian.

Emilie Dufresne (University of Nottingham): Separating invariants and local cohomology The study of separating invariants is a new trend in Invariant Theory and a return to its roots: invariants as a classification tool. For a finite group acting linearly on a vector space, a separating set is simply a set of invariants whose elements separate the orbits o the action. Such a set need not generate the ring of invariants. In this talk, we give lower bounds on the size of separating sets based on the geometry of the action. These results are obtained via the study of the local cohomology with support at an arrangement of linear subspaces naturally arising from the action. (Joint with Jack Jeffries)

Xiaoting Zhang (Uppsala Universitet): Pyramids and their applications In this talk, we define the category of pyramids over an additive category. If a strict monoidal structure is enclosed over the additive category, the strictness will be preserved when the monidal structure is lifted to the category of pyramids (avoiding any use of direct sums). The latter is biequivalent to the category of complexes and this biequivalence can be induced onto the corresponding homotopy categories. And a strict monoidal action can be also defined on the (homotopy) category of pyramids. As an application, we prove that every simple transitive 2-representation of the 2-category of projective bimodules over a finite dimensional algebra is equivalent to a cell 2-representation. This is joint work with Volodymyr Mazorchuk and Vanessa Miemietz.

Sue Sierra (University of Edinburgh): Chain conditions in the enveloping algebra of the Witt algebra The Witt algebra W is the Lie algebra of vector fields on the complex torus. In 2013 Sierra and Walton answered a long-standing question by showing that the enveloping algebras of the Witt algebra and of its positive part are neither left or right noetherian. In particular, the kernel of the natural map to the (localised) Weyl algebra is not finitely generated as a left or right ideal, although as a two-sided ideal it is principal. Furthermore, U(W) has infinite Gelfand-Kirillov dimension, whereas subalgebras of the Weyl algebra have polynomial growth.
This suggests two conjectures: that these enveloping algebras satisfy the ascending chain condition on two-sided ideals, and that any proper factor algebra has finite GK-dimension. We present work in progress on both conjectures. We show that the symmetric algebra S(W) has ACC on Poisson radical ideals. We also show that any factor of S(W) by an order 2 element has finite GK-dimension. This is joint work with Alexey Petukhov.

Ana Garcia Elsener (Universität Graz): Triangles in cluster categories and punctured skein relations We describe triangles in the cluster category of type D, and show the relation between them and punctured skein relations arising from Teichmuler theory.

Joseph Karmazyn (University of Sheffield): Threefold flops, matrix factorisations, and noncommutative algebras. Threefold flops in algebraic geometry were classified into 6 families by Katz and Morrison using the length invariant. Curto and Morrison have also given an efficient and explicit description of the first two families via matrix factorisations, and they conjecture that this can be done generally.
I plan to recap and discuss these results and to explain how the classification can be explicitly realised using noncommutative algebra.

Diane Maclagan (University of Warwick): Tropical ideals. Tropical geometry studies geometry over the tropical semiring, where multiplication is replaced by addition and addition by minimum. Over the last fifteen years there has been an explosion of work on varieties in this setting.
From a commutative algebra perspective, however, the semiring of tropical polynomials is not as nice as the its standard counterpart. We no longer have unique factorization, cancellation, or the Noetherian property. I will discuss joint work with Felipe Rincon on a special class of "tropical" ideals in this semiring that is much better behaved, and on the geometric side allows us to expand from varieties to schemes. This uses the theory of valuated matroids.

Idan Eisner (University of Loughborough): Exotic cluster algebras on simple Lie groups. Using the notion of compatibility between Poisson brackets and cluster algebras in the coordinate rings of simple complex Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified by the Belavin-Drinfeld classification of solutions to the classical Yang Baxter equation. For a simple complex Lie group G and a Belavin-Drinfeld class, one can define a corresponding Poisson bracket on the ring of regular functions on G. For some of these classes a compatible cluster structure can be constructed. We will describe some of these for G=SLn. In some cases, the compatible structure is a generalized cluster algebra, where the exchange relations are polynomial rather than binomial. We will see an example for G=SP6.

Sira Gratz (University of Glasgow): Noncrossing partitions and thick subcategories. Ingalls and Thomas have shown that the lattice of non-crossing partitions of a regular polygon with $n+1$ vertices is isomorphic to the lattice of thick subcategories in the bounded derived category of representations of a Dynkin quiver of type A with n vertices. In joint work with Greg Stevenson we provide an infinite version of this result by showing that the lattice of non-crossing partitions of the infinity-gon with a point at infinity is isomorphic to the lattice of thick subcategories in the bounded derived category of graded modules over the dual numbers.

Jordan McMahon (Universität Graz): Higher Frieze Patterns. Frieze patterns have an interesting combinatorial structure, as studied by Conway and Coxeter. This structure was later found to illustrate many of the properties of basic cluster algebras, and several generalisations of frieze patterns have since appeared in the literature and expanded on this connection. One such generalisation, the higher frieze pattern, introduces a link to higher Auslander-Reiten theory. We will describe higher frieze patterns, and how to generalise some of the classical combinatorial properties such as the quiddity sequence.

Philipp Lampe (Durham University): Factoriality and class groups of cluster algebras. We report on a joint work with Ana Garcia Elsener and Daniel Smertnig about the factorization theory of cluster algebras. The talk begins with an introduction to class groups and cluster algebras. Next, we give an explicit description of class groups of acyclic cluster algebras in terms of partner sets. As a Krull domain is factorial if and only its class group is zero, this constitutes a generalization of the study of factoriality of cluster algebras started by Geiß-Leclerc-Schröer. As a corollary, we reprove and extend a classification of factoriality for cluster algebras of Dynkin type.

Michael Wemyss (University of Glasgow): Intersections in Tits Cones, and Applications. I will begin by giving a purely combinatorial construction of a 2-sphere, with some points missing, based only on some combinatorial information about Dynkin diagrams. This punctured sphere corresponds to the physicists' stringy Kahler moduli space for a certain 3-dimensional surgery in algebraic geometry, and it gives us various predictions for the derived symmetry group, which is why we care. However, the talk is completely algebraic, and mostly combinatorial, and describes how to produce "affine" hyperplane arrangements for various Coxeter arrangements/groups which traditionally don't have affine versions. Everything is encoded by choices of nodes in Dynkin diagrams. At the end, I will briefly explain some applications to noncommutative resolutions, to tilting theory, and to group actions.

Lutz Hille (Universität Münster): Exceptional and spherical modules for the Auslander algebra of $k[T]/T^t$ (jt. w. David Ploog). The Auslander algebra of the truncated polynomial ring $k[T]/T^t$ plays an important role for many problems in algebra and geometry. It is a tilted algebra for some configurations of (-2)-curves, it is related to actions of parabolic groups, it is related to the Springer resolution, and it contains many spherical modules. Thus it is desirable to solve certain classification problems for this algebra.
In this talk we classify all exceptional modules, full exceptional sequences and spherical modules and have a look towards the classification of those objects in the derived category.

Brent Pym (University of Edinburgh): Holonomic Poisson manifolds and deformations of elliptic algebras. Determining the moduli space of deformations of a given Poisson bracket is typically a difficult problem. The deformation space is usually infinite-dimensional, highly obstructed, and extremely sensitive to singularities of the bracket. I will describe joint work with Travis Schedler, in which we introduce a natural new nondegeneracy condition for Poisson brackets, called holonomicity. It ensures strong finiteness properties for the relevant deformation complex, making the deformation spaces computable in terms of topological invariants such as intersection cohomology. As an application, we establish the deformation-invariance of some families of noncommutative algebras introduced by Sklyanin and Feigin--Odesskii.

Arnaud Mortier (Dublin City University): Kontsevich invariants in knot theory. We will investigate different aspects of the Kontsevich integral, a strong knot invariant with roots in mathematical physics, algebraically related to the study of knots via singularity theory. We will see how to define a $1$-cocycle using the same ideas. I will explain the grounds of knot theory, what a 1-cocycle is in this context, and what we can expect from such a tool.

Raphael Bennett-Tennenhaus (University of Leeds): Some generalisations of special biserial algebras. In this talk I shall construct generalisations of the special biserial algebras introduced and studied by Pogorzaly and Skowronski. I will then survey some results from my PhD thesis, in which these generalisations were the focus. The talk shall be relatively self contained.

Hendrik Suess (University of Manchester): Frobenius splitting of toric and T-varieties . In positive characteristic some standard tools of algebraic geometry stop working, such as Kodaira vanishing or resolution of singularities, but as a compensation we also obtain a very powerful new tool: The Frobenius morphism. For example, if there exists a section of this morphism we get back the Kodaira vanishing theorem. Such sections are called Frobenius splittings. Following a paper of Sam Payne I am going to discuss Frobenius splittings on toric varieties and connect them to the combinatorics of lattice polytopes. Then I will present a generalisation of Payne's results to so-called T-varieties. This is joint work with Piotr Achinger and Nathan Ilten.

Amit Hazi (University of Leeds): $p$-adic expansions of affine type Soergel bimodules. The diagrammatic category of Soergel bimodules is a linear, additive, monoidal category with deep connections to Kazhdan-Lusztig theory and representation theory. In this talk I will introduce a functor defined over positive characteristic Soergel bimodules for an affine Weyl group, which in some sense is akin to base $p$ or $p$-adic expansion of ordinary integers.

David Pauksztello (Lancaster University): Silting theory and stability spaces. In this talk I will introduce the notion of silting objects and mutation of silting objects. I will then show how the combinatorics of silting mutation can give one information regarding the structure of the space of stability conditions. In particular, I will show how a certain discreteness of this mutation theory enables one to employ techniques of Qiu and Woolf to obtain the contractibility of the space of stability conditions for a class of mainstream algebraic examples, the so-called silting-discrete algebras. This talk will be a discussion of joint work with Nathan Broomhead, David Ploog, Manuel Saorin and Alexandra Zvonareva.

Alice Rizzardo (University of Liverpool): Enhancements in derived and triangulated categories . Derived and triangulated categories are a fundamental object of study for many mathematicians, both in geometry and in topology. Their structure is however in many ways insufficient, and usually an enhancement is needed to carry on many important constructions on them. In this talk we will discuss existence and uniqueness of such enhancements for triangulated categories defined over a field.

Colin Ingalls (Carleton University): McKay quivers. Fix a finite group $G$ and a representation $W$, the McKay quiver has vertices given by irreducible representations $V_i$ and $\dim\mathrm{Hom}_G(W\otimes V_i,V_j)$ many arrows between $V_i$ and $V_j$. We briefly present the history of McKay quivers and their applications in geometry and representation theory. Then we discuss recent descriptions of McKay quivers of reflection groups by M. Lewis, and work with E. Faber and R. Buchweitz applying results of Lusztig on McKay quivers to understand the relations of the basic model of the skew group ring.