Tame categories, geometric models, and homological mirror symmetry
Paderborn University, Lecture hall A3
Speakers:
Participants:
Schedule:
| Monday | Tuesday | Thursday | Friday | ||
|---|---|---|---|---|---|
| 11.00 – 11.50 | Sibylle Schroll | ||||
| 12.00 – 14.00 | Lunch break | ||||
| 14.00 – 14.50 | Raf Bocklandt | Sebastian Opper | Alexandra Zvonareva | Christof Geiß | |
| 15.00 – 15.50 | Daniel Labardini-Fragoso | Cheol-Hyun Cho | Raphael Bennett-Tennenhaus | ||
| 16.00 – 16.30 | Coffee break | Coffee break | |||
| 16.30 – 17.20 | Karin Baur | Merlin Christ | Jasper van de Kreeke | ||
Organisers: Igor Burban, William Crawley-Boevey, Henning Krause, Kyungmin Rho
Abstracts:
Karin Baur: Grassmannians cluster categories
We study the category of Cohen-Macaulay modules over a quotient of a preprojective algebra. These categorify the cluster algebra structure of the coordinate ring of the Grassmannian. In particular, we study the tame cases and the Auslander-Reiten quiver of these categories. We relate rigid indecomposable modules with real roots for the corresponding Kac Moody algebra.
We study the category of Cohen-Macaulay modules over a quotient of a preprojective algebra. These categorify the cluster algebra structure of the coordinate ring of the Grassmannian. In particular, we study the tame cases and the Auslander-Reiten quiver of these categories. We relate rigid indecomposable modules with real roots for the corresponding Kac Moody algebra.
Raphael Bennett-Tennenhaus: Semilinear clannish algebras
Clannish algebras are certain quotients of path algebras introduced by Crawley-Boevey. Each relation is either a path, or a quadratic polynomial in a loop that factors with distinct roots over the ground field. I will discuss a larger class of rings. On the one hand, the more general notion of a semilinear clannish algebra retains the property of having a tame module category. On the other hand, the definition specifies to interesting examples of rings. By permitting irreducible quadratics, we recover representations arising in work of Geuenich and Labardini-Fragoso. By allowing arrows to skew scalars by automorphisms, we recover representations arising in work of Kottwitz and Rapoport. This talk is based on joint work with Crawley-Boevey (2204.12138) and joint work with Labardini-Fragoso (2303.05326).
Clannish algebras are certain quotients of path algebras introduced by Crawley-Boevey. Each relation is either a path, or a quadratic polynomial in a loop that factors with distinct roots over the ground field. I will discuss a larger class of rings. On the one hand, the more general notion of a semilinear clannish algebra retains the property of having a tame module category. On the other hand, the definition specifies to interesting examples of rings. By permitting irreducible quadratics, we recover representations arising in work of Geuenich and Labardini-Fragoso. By allowing arrows to skew scalars by automorphisms, we recover representations arising in work of Kottwitz and Rapoport. This talk is based on joint work with Crawley-Boevey (2204.12138) and joint work with Labardini-Fragoso (2303.05326).
Raf Bocklandt: Deformations of Gentle $A_\infty$-Algebras
Gentle $A_\infty$-algebras can be used to describe fukaya categories of punctured surfaces and their deformation theory corresponds to filling these punctures with (orbifold) points. In this talk we will discuss in detail how this works. In particular we will look at the Hochschild cohomology of gentle $A_\infty$-algebras of arc collections on marked surfaces without boundary components and construct explicit curved deformations of these algebras. When the underlying arc collection has no loops or two-cycles, we show that the dgla structure of the Hochschild complex is formal and give an explicit realization of all deformations up to gauge equivalence. Finally we discuss how to transfer these deformations to the category of twisted complexes. This is joint work with Jasper van de Kreeke.
Gentle $A_\infty$-algebras can be used to describe fukaya categories of punctured surfaces and their deformation theory corresponds to filling these punctures with (orbifold) points. In this talk we will discuss in detail how this works. In particular we will look at the Hochschild cohomology of gentle $A_\infty$-algebras of arc collections on marked surfaces without boundary components and construct explicit curved deformations of these algebras. When the underlying arc collection has no loops or two-cycles, we show that the dgla structure of the Hochschild complex is formal and give an explicit realization of all deformations up to gauge equivalence. Finally we discuss how to transfer these deformations to the category of twisted complexes. This is joint work with Jasper van de Kreeke.
Cheol-Hyun Cho: Topological Fukaya category of tagged arcs
A tagged arc on a surface is introduced by Fomin, Shapiro, and Thurston to study cluster theory on marked surfaces. Given a tagged arc system on a graded marked surface, we define its $\mathbb{Z}$-graded $A_\infty$-category, generalizing the construction of Haiden, Katzarkov, and Kontsevich for arc systems. When a tagged arc system arises from a non-trivial involution on a marked surface, we show that this $A_\infty$-category is quasi-isomorphic to the invariant part of the topological Fukaya category under the involution. In particular, this identifies tagged arcs with non-geometric idempotents of Fukaya category. This is a joint work with Kyoungmo Kim.
A tagged arc on a surface is introduced by Fomin, Shapiro, and Thurston to study cluster theory on marked surfaces. Given a tagged arc system on a graded marked surface, we define its $\mathbb{Z}$-graded $A_\infty$-category, generalizing the construction of Haiden, Katzarkov, and Kontsevich for arc systems. When a tagged arc system arises from a non-trivial involution on a marked surface, we show that this $A_\infty$-category is quasi-isomorphic to the invariant part of the topological Fukaya category under the involution. In particular, this identifies tagged arcs with non-geometric idempotents of Fukaya category. This is a joint work with Kyoungmo Kim.
Merlin Christ: $3$-Calabi-Yau categories arising from surfaces: gluing and geometric models
Labardini-Fragoso associated with each oriented marked surface a quiver with potential, which in turn defines a $3$-Calabi-Yau dg-category. These categories categories are considered for instance in Bridgland--Smith's work on stability conditions and in the categorification of cluster algebras of surfaces. In this talk, we will discuss different generalizations of these dg-categories that can also be associated with (decorated) surfaces. They all share the feature that they satisfy gluing (over for instance a triangulation of the surface), which can be formulated in terms of (co)sheaves of categories. These gluing properties allow to construct partial geometric models. This talk is partially based on joint work with Fabian Haiden and Qiu Yu.
Labardini-Fragoso associated with each oriented marked surface a quiver with potential, which in turn defines a $3$-Calabi-Yau dg-category. These categories categories are considered for instance in Bridgland--Smith's work on stability conditions and in the categorification of cluster algebras of surfaces. In this talk, we will discuss different generalizations of these dg-categories that can also be associated with (decorated) surfaces. They all share the feature that they satisfy gluing (over for instance a triangulation of the surface), which can be formulated in terms of (co)sheaves of categories. These gluing properties allow to construct partial geometric models. This talk is partially based on joint work with Fabian Haiden and Qiu Yu.
Christof Geiß: Homomorphisms between representations and generically $\tau$-reduced components for skewed-gentle algebras
Let $K$ be an algebraically closed field with characteristic different from $2$, and $A$ a skewed-gentle $K$-algebra. In this case, Crawley-Boevey's description of the indecomposable $A$-modules becomes particularly easy. This allows us to provide an explicit basis for the homomorphisms between any two indecomposable representations in terms of the corresponding admissible words in the sense of Qiu and Zhou. Previously (Geiss, 1999), such a basis was only available when no asymmetric band modules were involved. We also extend a relaxed version of fringing and kisses from Brüstle et al. (2020) to the setting of skewed-gentle algebras. With this at hand, we obtain convenient formulae for the $E$-invariant and $g$-vectors in this context, which are similar to the known expressions for gentle algebras. Note however, that we allow in our setting also band modules. As an application, we give a description of the indecomposable, generically $\tau$-reduced irreducible components of the representation varieties of $A$ as well as the generic values of the $E$-invariant between them in terms tagged admissible words. This is useful for topological models.
Let $K$ be an algebraically closed field with characteristic different from $2$, and $A$ a skewed-gentle $K$-algebra. In this case, Crawley-Boevey's description of the indecomposable $A$-modules becomes particularly easy. This allows us to provide an explicit basis for the homomorphisms between any two indecomposable representations in terms of the corresponding admissible words in the sense of Qiu and Zhou. Previously (Geiss, 1999), such a basis was only available when no asymmetric band modules were involved. We also extend a relaxed version of fringing and kisses from Brüstle et al. (2020) to the setting of skewed-gentle algebras. With this at hand, we obtain convenient formulae for the $E$-invariant and $g$-vectors in this context, which are similar to the known expressions for gentle algebras. Note however, that we allow in our setting also band modules. As an application, we give a description of the indecomposable, generically $\tau$-reduced irreducible components of the representation varieties of $A$ as well as the generic values of the $E$-invariant between them in terms tagged admissible words. This is useful for topological models.
Daniel Labardini-Fragoso: (Generalized) cluster algebras from surfaces with orbifold points via gentle algebras
To a surface with marked points on the boundary and orbifold points one can associate a skew-symmetrizable cluster algebra following Felikson--Shapiro--Tumarkin, or a generalized cluster algebra following Chekhov--Shapiro. To each triangulation of the surface one can associate a gentle algebra, and to each arc on the surface a representation of this gentle algebra. In this talk I will present joint work with Lang Mou in which we show that the locally free Caldero--Chapoton function of this representation is the cluster variable in the skew-symmetrizable cluster algebra, whereas the usual Caldero--Chapoton function is the cluster variable in the generalized cluster algebra.
To a surface with marked points on the boundary and orbifold points one can associate a skew-symmetrizable cluster algebra following Felikson--Shapiro--Tumarkin, or a generalized cluster algebra following Chekhov--Shapiro. To each triangulation of the surface one can associate a gentle algebra, and to each arc on the surface a representation of this gentle algebra. In this talk I will present joint work with Lang Mou in which we show that the locally free Caldero--Chapoton function of this representation is the cluster variable in the skew-symmetrizable cluster algebra, whereas the usual Caldero--Chapoton function is the cluster variable in the generalized cluster algebra.
Sebastian Opper: Derived Picard groups and integration of Hochschild cohomology
I will talk about a general tool which allows one to study derived Picard groups, an enhanced version of symmetry groups for triangulated categories. These groups are rather elusive and their description requires a very good understanding of the triangulated category at hand. Some cases where they have been successfully computed by ad-hoc methods include the derived categories of hereditary and canonical algebras, local and commutative rings as well as Brauer tree and derived discrete algebras. A result of Keller shows that the Lie algebra of the derived Picard group of an algebra can be identified with its Hochschild cohomology equipped with the Gerstenhaber Lie bracket. In analogy with the classical relationship between Lie groups and their Lie algebras, I will explain how to "integrate" elements in the Hochschild cohomology of an $A_\infty$-category over fields of characteristic zero to elements in the derived Picard group via a generalized exponential map. Afterwards we discuss properties of this exponential and a few applications. The first concerns necessary conditions for the uniqueness of lifts of functors from homotopy categories to enhancements and the related uniqueness problem of Fourier-Mukai kernels. Other applications concern derived Picard groups of categories arising in algebra and geometry which we illustrate by Fukaya categories of cotangent bundles and their plumbings as well as derived categories of graded gentle algebras and the corresponding partially wrapped Fukaya categories of surfaces in the sense of Haiden-Katzarkov-Kontsevich. Based on preprint arXiv:2405.14448.
I will talk about a general tool which allows one to study derived Picard groups, an enhanced version of symmetry groups for triangulated categories. These groups are rather elusive and their description requires a very good understanding of the triangulated category at hand. Some cases where they have been successfully computed by ad-hoc methods include the derived categories of hereditary and canonical algebras, local and commutative rings as well as Brauer tree and derived discrete algebras. A result of Keller shows that the Lie algebra of the derived Picard group of an algebra can be identified with its Hochschild cohomology equipped with the Gerstenhaber Lie bracket. In analogy with the classical relationship between Lie groups and their Lie algebras, I will explain how to "integrate" elements in the Hochschild cohomology of an $A_\infty$-category over fields of characteristic zero to elements in the derived Picard group via a generalized exponential map. Afterwards we discuss properties of this exponential and a few applications. The first concerns necessary conditions for the uniqueness of lifts of functors from homotopy categories to enhancements and the related uniqueness problem of Fourier-Mukai kernels. Other applications concern derived Picard groups of categories arising in algebra and geometry which we illustrate by Fukaya categories of cotangent bundles and their plumbings as well as derived categories of graded gentle algebras and the corresponding partially wrapped Fukaya categories of surfaces in the sense of Haiden-Katzarkov-Kontsevich. Based on preprint arXiv:2405.14448.
Sibylle Schroll: Partially wrapped Fukaya categories of orbifold surfaces
Motivated by the consideration of $A_\infty$-deformations of gentle algebras, we define partially wrapped Fukaya categories of surfaces with orbifold points by giving several equivalent constructions. In particular, this gives rise to geometric realisations of the perfect derived categories of skew-gentle algebras in terms of dissections of the orbifold surfaces into polygons. By considering a wider class of dissections, we are able to identify all dissections giving rise to formal $A_\infty$-algebras and thus a large class of associative algebras derived equivalent to skew-gentle algebras. This raises the question whether this is a complete description of the derived equivalence classes of skew-gentle algebras.
This is joint work with Severin Barmeier and Zhengfang Wang.
Motivated by the consideration of $A_\infty$-deformations of gentle algebras, we define partially wrapped Fukaya categories of surfaces with orbifold points by giving several equivalent constructions. In particular, this gives rise to geometric realisations of the perfect derived categories of skew-gentle algebras in terms of dissections of the orbifold surfaces into polygons. By considering a wider class of dissections, we are able to identify all dissections giving rise to formal $A_\infty$-algebras and thus a large class of associative algebras derived equivalent to skew-gentle algebras. This raises the question whether this is a complete description of the derived equivalence classes of skew-gentle algebras.
This is joint work with Severin Barmeier and Zhengfang Wang.
Jasper van de Kreeke: Resolutions of Kleinian singularities via Tannaka duality
About 30 years ago, physicists constructed a bundle whose fibers are resolutions of Kleinian singularities, with two notable fibers: the orbifold resolution and the minimal resolution. In modern terms, the base of the bundle consists of Bridgeland stability conditions. Mysteriously, the orbifold resolution is associated with an imaginary stability condition and is therefore missed by classical GIT. This year, Abdelgadir and Segal revisited this mystery in the Kleinian $D_4$ case and constructed a variety which simultaneously produces both resolutions as GIT quotients. Their approach effectively stretches the stability space, shifting the imaginary value to the real axis. In this talk, I show how to extend their method to all Kleinian singularities.
About 30 years ago, physicists constructed a bundle whose fibers are resolutions of Kleinian singularities, with two notable fibers: the orbifold resolution and the minimal resolution. In modern terms, the base of the bundle consists of Bridgeland stability conditions. Mysteriously, the orbifold resolution is associated with an imaginary stability condition and is therefore missed by classical GIT. This year, Abdelgadir and Segal revisited this mystery in the Kleinian $D_4$ case and constructed a variety which simultaneously produces both resolutions as GIT quotients. Their approach effectively stretches the stability space, shifting the imaginary value to the real axis. In this talk, I show how to extend their method to all Kleinian singularities.
Alexandra Zvonareva: Derived categories over Brauer graph algebras and the action of the mapping class group
Graded Brauer graph algebras can be defined from a minimal embedding of a graph into a compact oriented surface with boundary which is additionally equipped with a line field. Under suitable assumptions these algebras are $n$-Calabi-Yau and projective modules over these algebras are $n$-spherical. Thus, the category of perfect complexes over graded Brauer graph algebras is a natural setting to study the actions of the groups generated by various configurations of spherical objects. In this talk I will discuss the action of certain subgroups of the mapping class group of the underlying surface on the category of perfect complexes over graded Brauer graph algebras. This is based on a joint work in progress with Wassilij Gnedin and Sebastian Opper.
Graded Brauer graph algebras can be defined from a minimal embedding of a graph into a compact oriented surface with boundary which is additionally equipped with a line field. Under suitable assumptions these algebras are $n$-Calabi-Yau and projective modules over these algebras are $n$-spherical. Thus, the category of perfect complexes over graded Brauer graph algebras is a natural setting to study the actions of the groups generated by various configurations of spherical objects. In this talk I will discuss the action of certain subgroups of the mapping class group of the underlying surface on the category of perfect complexes over graded Brauer graph algebras. This is based on a joint work in progress with Wassilij Gnedin and Sebastian Opper.