Let $Λ_Q$ be the preprojective algebra of a finite acyclic quiver $Q$ of non-Dynkin type and $D^b(\mathrm{rep}^n Λ_Q)$ be the bounded derived category of finite dimensional nilpotent $Λ_Q$-modules. We define spherical twist functors over the root category $\mathcal{R}_{Λ_Q}$ of $D^b(\mathrm{rep}^n Λ_Q)$ and then realize the Weyl group associated to $Q$ as certain subquotient of the automorphism group of the Ringel-Hall Lie algebra $\mathfrak{g}(\mathcal{R}_{Λ_Q})$ of $\mathcal{R}_{Λ_Q}$ induced by spherical twist functors. We also present a conjectural relation between certain Lie subalgebras of $\mathfrak{g}(\mathcal{R}_{Λ_Q})$ and $\mathfrak{g}(\mathcal{R}_Q)$, where $\mathfrak{g}(\mathcal{R}_Q)$ is the Ringe-Hall Lie algebra associated to the root category $\mathcal{R}_Q$ of $Q$.
Deformed preprojective algebras are generalizations of the usual preprojective algebras introduced by Crawley-Boevey and Holland, which have applications to Kleinian singularities, the Deligne-Simpson problem, integrable systems and noncommutative geometry. In this paper we offer three contributions to the study of such algebras: (1) the 2-Calabi-Yau property; (2) the unification of the reflection functors of Crawley-Boevey and Holland with reflection functors for the usual preprojective algebras; and (3) the classification of tilting ideals in 2-Calabi-Yau algebras, and especially in deformed…
In this article we study the properties of preprojective algebras of representation finite species. To understand the structure of a preprojective algebra, one often studies its Nakayama automorphism. A complete description of the Nakayama automorphism is given by Brenner, Butler and King when the algebra is given by a path algebra. We generalize this result to the species case. We show that the preprojective algebra of a representation finite species is an almost Koszul algebra. With this we know that almost Koszul complexes exist. It turns out that the almost Koszul complex for a representat…
We introduce a very general extension of the monomorphism category as studied by Ringel and Schmidmeier which in particular covers generalised species over locally bounded quivers. We prove that analogues of the kernel and cokernel functor send almost split sequences over the path algebra and the preprojective algebra to split or almost split sequences in the monomorphism category. We derive this from a general result on preservation of almost split morphisms under adjoint functors whose counit is a monomorphism. Despite of its generality, our monomorphism categories still allow for explicit c…
We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a gr…
For a semisimple complex Lie algebra $\mathfrak g$, the BGG category $\mathcal{O}$ is of particular interest in representation theory. It is known that Irving's shuffling functors $\mathrm{Sh}_{w}$, indexed by elements $w\in W$ of the Weyl group, induce an action of the braid group $B_W$ associated to $W$ on the derived categories $D^\mathrm{b}(\mathcal{O}_λ)$ of blocks of $\mathcal{O}$. We show that for maximal parabolic subalgebras $\mathfrak{p}$ of $\mathfrak{sl}_n$ corresponding to the parabolic subgroup $W_\mathfrak{p}=S_{n-1}\times S_1$ of $S_n$, the derived shuffling functors $\mathbf{L…
We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor $Σ$ on a finite abelian category $\mathcal{M}$, we introduce the notion of a $Σ$-twisted trace on the class $\mathrm{Proj}(\mathcal{M})$ of projective objects of $\mathcal{M}$. In our framework, there is a one-to-one correspondence between the set of $Σ$-twisted traces on $\mathrm{Proj}(\mathcal{M})$ and the set of natural transformations from $Σ$ to the Nakayama functor of $\mathcal{M}$. Non-degeneracy and compatibility with the module structure (when $\mat…
We develop a theory of Frobenius functors for symmetric tensor categories (STC) $\mathcal{C}$ over a field $\bf k$ of characteristic $p$, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor $F: \mathcal{C}\to \mathcal{C}\boxtimes {\rm Ver}_p$, where ${\rm Ver}_p$ is the Verlinde category (the semisimplification of ${\rm Rep}_{\bf k}(\mathbb{Z}/p)$). This generalizes the usual Frobenius twist functor in modular representation theory and also one defined in arXiv:1503.01492, where it is used to show that if $\mathcal{C}$ i…
In this paper, we give a Verlinde formula for computing the ranks of the bundles of twisted conformal blocks associated with a simple Lie algebra equipped with an action of a finite group $Γ$ and a positive integral level $\ell$ under the assumption that "$Γ$ preserves a Borel". As a motivation for this Verlinde formula, we prove a categorical Verlinde formula which computes the fusion coefficients for any $Γ$-crossed modular fusion category as defined by Turaev. To relate these two versions of the Verlinde formula, we formulate the notion of a $Γ$-crossed modular functor and show that it is v…
We extend Colmez's functor defined for $\operatorname{GL}_2(\mathbf{Q}_p)$ to the category of finitely generated smooth admissible mod-$p$ representations of the two-fold metaplectic cover of $\operatorname{GL}_2(\mathbf{Q}_p)$. We compute the images of the absolutely irreducible genuine objects and obtain a bijection between the genuine supersingular representations and four-dimensional irreducible Galois representations invariant under twist by all characters of order two. Restricted to genuine objects, the extended functor naturally takes values in the category of what we call metaplectic G…
We adapt a construction due to Troesch to the category of strict polynomial superfunctors in order to construct complexes of injective objects whose cohomology is isomorphic to Frobenius twists of the (super)symmetric power functors. We apply these complexes to construct injective resolutions of the even and odd Frobenius twist functors, to investigate the structure of the Yoneda algebra of the Frobenius twist functor, and to compute other extension groups between strict polynomial superfunctors.
We prove that the action of a generalized braid group on an enhanced triangulated categories, generated by spherical twist functors along an ADE-configuration of $ω$-spherical objects, is faithful for any integer $ω\neq 1$.
Fractionally Calabi-Yau algebras and cluster tilting
Graduate School of Mathematics, Nagoya University（名古屋大学数学研究科）；Graduate School of Mathematical Sciences, University of Tokyo（东京大学数理科学研究科）；Mathematical Institute of the University of Bonn（波恩大学数学研究所）
We build a bijection between the set $\sttiltΛ$ of isomorphism classes of basic support $τ$-tilting modules over the Auslander algebra $Λ$ of $K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is an anti-isomorphism of partially ordered sets with respect to the generation order on $\sttiltΛ$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set $\tiltΛ$ of isomorphism classes of basic tilting $Λ$-modules and the symmetric group $\mathfrak{S}_n$ due to Brüstle, Hille, Ringel and Röhrle. Regarding the preprojective algebra $Γ$ of Dynkin type $A…
