Speaker: Bernhard Keller / 孔博恩 (Universite Paris Cite / 巴黎西岱大学)
Time: 2025-12-22 15:00-18:00
Location: 1110, SIMIS
摘要：
Higgs categories are certain exact dg categories which
serve to (additively) categorify cluster algebras with (non invertible)
coefficients. They were introduced by Yilin Wu in 2023. A basic example
is the category of finite-dimensional modules over the preprojective algebra of a
Dynkin quiver (i.e. a quiver whose underlying graph is an ADE Dynkin diagram).
As shown by Geiss-Leclerc-Schroer (2006), this category allows to categorify the
coordinate algebra of the maximal unipotent subgroup in the simple algebraic
group of the same Dynkin type. While in this basic example, the Higgs
category is a Quillen exact category concentrated in degree 0, the
Higgs categories corresponding to cluster varieties like double
Bruhat cells or triples of flags have homologies in infinitely
many non zero degrees. In this talk, we will introduce Higgs categories and
illustrate the construction on many examples. We will then report on
recent joint work with Miantao Liu (based on previous work by Yilin Wu,
by Xiaofa Chen and by Zhenhui Ding), where we use Gorenstein projective
dg modules to describe Higgs categories. This allows to prove an
equivalence conjectured by Merlin Christ and to lift Goncharov-Shen’s
symmetries of varieties of triples of flags to the categorical level.
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