Post-workshop Seminar on "Algebraic Triangulated Categories and Related Topics"

26 July (Sunday), Room I-307

11:00-12:00 Hiroyuki Minamoto (Kyoto University)
Ampleness of two-sided tilting complexes and Fano algebras

In this talk we introduce a notion of ampleness for two-sided tilting
complexes over finite dimensional algebras of finite global dimension,
and show its basic properties, which justify the name "ampleness".
We define Fano algebras by the anti-ampleness of shifted Serre functor.
Fano algebras have remarkable properties. Some classes of algebras
studied before are Fano. There is good supply of new Fano algebras.
We can construct Fano algebras from AS-regular algebras (without
finiteness of Gelfand-Kirillov dimension). As an application, we prove
that AS-regular algebras are graded coherent.

13:30--14:30 David Ploog (Berlin)
Comparing categorical lifts of certain root lattices

Abstract: This is a report about work in progress (with Chris Brav). We
present a setting in which full exceptional collections and spherical
collections can be compared. This allows to relate two different ways in
which the ADE root lattices (coming from ADE singularities) can be lifted
to triangulated categories: via graded matrix factorisations (following
Kajiura, Saito, Takahashi, Ueda) or via sheaves on the resolution.

15:00--16:00 Atsushi Takahashi (Osaka University)

16:30--17:30 Bernhard Keller (Paris 7)
Derived equivalences of deformed Calabi-Yau completions

Abstract: We recall the construction of deformed Calabi-Yau completions and the
Ginzburg algebra. We use this framework to give a conceptual proof of the derived
invariance theorem for Ginzburg algebras, which was first obtained in joint work
with Dong Yang.
27 July (Monday)

10:00--11:00, Room I-307, Martin Herschend (Nagoya University)
The Clebsch-Gordan problem for string algebras

String algebras are a class of tame algebras whose indecomposable finite
dimensional modules are classified by strings and bands. They originate from
the classification of Harish-Chandra modules over the Lorentz group by
Gelfand and Ponomarev. By definition, any string algebra is defined as the
path algebra of a quiver subject to certain monomial relations. Hence its
module category is equipped with a tensor product defined point-wise and
arrow-wise in terms of the underlying quiver. This leads to the following
problem: given two indecomposable modules over a string algebra, decompose
their tensor product into a direct sum of indecomposables. In my talk I will
present the solution to this problem for all string algebras. Moreover, I
will describe the corresponding representation rings.

11:30--12:30, Room I-307, Yoshiyuki Kimura (Kyoto University)
Affine quivers and the crystal $B(\infty)$

This talk is based on a joint work with Hiraku Nakajima.
We study a relationship between the two geometric construction of the
crystal $B(\infty)$. One is the Lusztig's canonical base which is described
as simple perverse sheaves and another is the Lagrangian construction by
using Lusztig's quiver varieties due to M.Kashiwara and Y.Saito.
More precisely, we give an estimate of characteristic variety of canonical
base in general affine quiver cases (these simple perverse sheaves were
descrived by G.Lusztig (Y.Li and Z.Lin) in terms of Dlab-Ringel's theory
of affine quivers). In particular, we introduce a partial order on
$B(\infty)$ which is motivated by the order on (affine) PBW base defined by
Beck-Chari-Pressley and Beck-Nakajima.

13:30--14:30, Room I-307, Tomoki Nakanishi (Nagoya University)
T-systems, Y-systems, and cluster algebras

The T-systems and Y-systems are systems of relations originated from
the study of integrable quantum systems in 90's and relevant to the
representation theory of the quantum groups. Initiated by
Fomin-Zelevinsky and Keller, the periodicity phenomena of the systems
recently attract attention from the point of views of cluster algebras
and cluster categories. In this talk I will given an introductory
review of what the T- and Y-systems are and how they are related to
cluster algebras.

15:00--16:00, Room I-552, Matthew Clarke (University of Cambridge)
Bismash products and group algebras

We will begin by introducing Hopf algebras together with the easiest
examples, group algebras and their duals, before giving a
rough overview of the classification attempt for finite-dimensional
complex semisimple Hopf algebras. We will then go on to describe more
interesting constructions, bicrossed and bismash products,
which can be defined using any finite group with a given factorisation into
trivially intersecting subgroups, and explain a way of studying
bismash products using character theory. We will examine a particular
family of bismash products arising from factorisations of the projective
general linear group over a finite field together with their
comultiplication twistings, which have been shown to be examples of so-called
non-trivial Hopf algebras. We also consider what happens if the factorised
group we started is a) a semidirect product, b) a Frobenius group.

16:30--17:30, Room I-552, Nanhua Xi (Chinese Academy of Sciences)
A new proof for classification of irreducible modules of a Hecke algebra of type $ A_{n-1}$

The classification of irreducible modules of a Hecke algebra of
type $A\_{n-1}$ was obtained by Dipper and James in 1986. In this talk
I will give a new proof for the classification. Essentially the idea
is due to Dipper and James, Murphy, but we use Kazhdan-Lusztig
theory and an affine Hecke algebra of type $\tilde A_{n-1}$ to prove
this result by a direct calculation.