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Mikhail Gorsky

Publications and source records attributed to Mikhail Gorsky.

At least 19 recordsLinked to original sources
arXiv2026

Relations between categorifications of higher-dimensional type $A$ cluster combinatorics

We consider three categories arising from the higher Auslander algebras of type $A$ in relation to $d$-dimensional cluster combinatorics: $d$-exact subcategory of the module category of $A^d_{n+1}$ generated by the $d$-cluster-tilting object, the $(d+2)$-angulated cluster category, and the $d$-almost positive subcategory of the derived category (the higher analogue of the category of two-term complexes of projectives). We show that the third one, introduced by the second-named author, is the $d$-exangulated quotient of the other two, introduced by Oppermann and Thomas, by the ideals generated by morphisms factoring through morphisms from injective to projective objects, thus providing an algebraic connection between the two models of Oppermann-Thomas. This is a $d$-exangulated version in type $A$ of a result of Br\"ustle and Yang and its interpretation by the first-named author together with Fang, Palu, Plamondon and Pressland. It also explains a well-known coincidence between the number of 2-term silting complexes in type $A_{n}$ and of tilting modules in type $A_{n+1}$ from the $0$-Auslander perspective. We expect this to serve as a prototypical example of the $0$-Auslander correspondence in higher homological algebra.

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arXiv2026

A Categorification of Subword Complexes and Its Hall Algebra

Bergeron and Ceballos defined a Hopf algebra structure on equivalence classes of subword complexes. We introduce a category of subword complexes, endow it with a proto-exact-like structure, and show that the corresponding dual Hall Hopf algebra is isomorphic to the algebra of Bergeron-Ceballos. We prove that the full subcategory of root-independent objects is proto-abelian in the sense of Dyckerhoff. We give a categorical lift of flips in subword complexes. We consider a version of a category of formal direct sums of subobjects for a root-independent subword complex and interpret it in terms of quivers. If the corresponding quiver is a tree, the category is endowed with a proto-exact structure. We show that its Hall algebra is isomorphic to the Hall algebra of the category of representations of this quiver over $\mathbb{F}_1$. Under certain conditions, a flip corresponds to changing a proto-exact structure while keeping the category the same up to isomorphism, which corresponds to a non-trivial automorphism of the Hall algebra. In type $A$, this leads to a realization of the nilpotent part of the universal enveloping algebra and its automorphisms.

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arXiv2026

Positroid Links and Braid varieties

We study braid varieties and their relation to open positroid varieties. We discuss four different types of braids associated to open positroid strata and show that their associated Legendrian links are all Legendrian isotopic. In particular, we prove that each open positroid stratum can be presented as the augmentation variety for four different Legendrian fronts described in terms of either permutations, juggling patterns, cyclic rank matrices or Le diagrams. We also relate braid varieties to open Richardson varieties and brick manifolds, showing that the latter provide projective compactifications of braid varieties, with normal crossing divisors at infinity.

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arXiv2025

Comparing cluster algebras on braid varieties

Braid varieties parametrize linear configurations of flags with transversality conditions dictated by positive braids. They include and generalize reduced double Bruhat cells, positroid varieties, open Bott-Samelson varieties, and Richardson varieties, among others. Recently, two cluster algebra structures were independently constructed in the coordinate rings of braid varieties: one using weaves and the other using Deodhar geometry. The main result of the article is that these two cluster algebras coincide. More generally, our comparative study matches the different concepts and results from each approach to the other, both on the combinatorial and algebraic geometric aspects.

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arXiv2025

Preorders on maximal chains: hyperplane arrangements, Cambrian lattices, and maximal green sequences

We study preorders on (equivalence classes of) maximal chains in the general context of polygonal lattices endowed with suitably nice edge labellings. We show that, given a quotient of polygonal lattices, such edge labellings descend to the quotient, and that there is an induced order-preserving surjective map on the preordered sets of equivalence classes of maximal chains. Under a natural condition ensuring that the domain is a poset, the map is a contraction of preordered sets. We apply this to lattices of regions of simplicial hyperplane arrangements, where the preorders are partial orders, in particular to finite Coxeter arrangements. For the latter, each choice of Coxeter element gives us a different partial order on the set of equivalence classes of maximal chains; these generalise certain reoriented higher Bruhat orders in dimension two. The maps of posets of maximal chains induced by Cambrian congruences generalise the map of Kapranov and Voevodsky from the higher Bruhat orders to the higher Stasheff--Tamari orders in dimension two. While the fibres of this map are known not always to be intervals, our results show that they are always connected. We show that, in the case of Cambrian lattices, the induced maps on maximal chains have nice descriptions in terms of orientations of rank-two root subsystems, in a way which resembles taking the stable objects of a Rudakov stability condition. We finally consider the algebraic realisation of the weak order and Cambrian lattices via torsion-free classes of preprojective and path algebras, relating the posets of maximal chains to our earlier work on maximal green sequences.

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arXiv2024

Counting in Calabi--Yau categories, with applications to Hall algebras and knot polynomials

We show that homotopy cardinality -- a priori ill-defined for many dg-categories, including all periodic ones -- has a reasonable definition for even-dimensional Calabi--Yau (evenCY) categories and their relative generalizations (under appropriate finiteness conditions). As a first application we solve the problem of defining an intrinsic Hall algebra for degreewise finite pre-triangulated dg-categories in the case of oddCY categories. We compare this definition with Toën's derived Hall algebras (in case they are well-defined) and with other approaches based on extended Hall algebras and central reduction, including a construction of Hall algebras associated with Calabi--Yau triples of triangulated categories. For a category equivalent to the root category of a 1CY abelian category $\mathcal A$, the algebra is shown to be isomorphic to the Drinfeld double of the twisted Ringel--Hall algebra of $\mathcal A$, thus resolving in the Calabi--Yau case the long-standing problem of realizing the latter as a Hall algebra intrinsically defined for such a triangulated category. Our second application is the proof of a conjecture of Ng--Rutherford--Shende--Sivek, which provides an intrinsic formula for the ruling polynomial of a Legendrian knot $L$, and its generalization to Legendrian tangles, in terms of the augmentation category of $L$.

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arXiv2024

Algebraic Weaves and Braid Varieties

In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting stratifications of braid varieties and their quotients. It is shown that the maximal charts of these stratifications are exponential Darboux charts for the holomorphic symplectic structures, and we relate these strata to exact Lagrangian fillings of Legendrian links.

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arXiv2024

Cluster structures on braid varieties

We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig's coordinates. Several explicit seeds are provided and the quiver and cluster variables are readily computable. We prove that these upper cluster algebras equal their cluster algebras, show local acyclicity, and explicitly determine their DT-transformations as the twist automorphisms of braid varieties. The main result also resolves the conjecture of B. Leclerc on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties.

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arXiv2024

Cluster deep loci and mirror symmetry

Affine cluster varieties are covered up to codimension 2 by open algebraic tori. We put forth a general conjecture (based on earlier conversation between Vivek Shende and the last author) characterizing their deep locus, i.e. the complement of all cluster charts, as the locus of points with non-trivial stabilizer under the action of cluster automorphisms. We use the diagrammatics of Demazure weaves to verify the conjecture for skew-symmetric cluster varieties of finite cluster type with arbitrary choice of frozens and for the top open positroid strata of Grassmannians $\mathrm{Gr}(2,n)$ and $\mathrm{Gr}(3,n)$. We illustrate how this already has applications in symplectic topology and mirror symmetry, by proving that the Fukaya category of Grassmannians $\mathrm{Gr}(2,2n+1)$ is split-generated by finitely many Lagrangian tori, and homological mirror symmetry holds with a Landau--Ginzburg model proposed by Rietsch. Finally, we study the geometry of the deep locus, and find that it can be singular and have several irreducible components of different dimensions, but they all are again cluster varieties in our examples in really full rank cases.

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arXiv2024

Extriangulated ideal quotients, with applications to cluster theory and gentle algebras

We extend results of Brüstle-Yang on ideal quotients of 2-term subcategories of perfect derived categories of non-positive dg algebras to a relative setting. We find a new interpretation of such quotients: they appear as prototypical examples of a new construction of quotients of extriangulated categories by ideals generated by morphisms from injectives to projectives. We apply our results to Frobenius exact cluster categories and Higgs categories with suitable relative extriangulated structures, and to categories of walks related to gentle algebras. In all three cases, the extriangulated structures are well-behaved (they are 0-Auslander) and their quotients are equivalent to homotopy categories of two-term complexes of projectives over suitable finite-dimensional algebras.

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arXiv2023

A functorial approach to rank functions on triangulated categories

We study rank functions on a triangulated category $\mathcal{C}$ via its abelianisation $\operatorname{mod}\mathcal{C}$. We prove that every rank function on $\mathcal{C}$ can be interpreted as an additive function on $\operatorname{mod}\mathcal{C}$. As a consequence, every integral rank function has a unique decomposition into irreducible ones. Furthermore, we relate integral rank functions to a number of important concepts in the functor category $\operatorname{Mod}\mathcal{C}$. We study the connection between rank functions and functors from $\mathcal{C}$ to locally finite triangulated categories, generalising results by Chuang and Lazarev. In the special case $\mathcal{C}=\mathcal{T}^c$ for a compactly generated triangulated category $\mathcal{T}$, this connection becomes particularly nice, providing a link between rank functions on $\mathcal{C}$ and smashing localisations of $\mathcal{T}$. In this context, any integral rank function can be described using the composition length with respect to certain endofinite objects in $\mathcal{T}$. Finally, if $\mathcal{C}=\operatorname{per}(A)$ for a differential graded algebra $A$, we classify homological epimorphisms $A\to B$ with $\operatorname{per}(B)$ locally finite via special rank functions which we call idempotent.

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arXiv2023

On the lattices of exact and weakly exact structures

We initiate in this article the study of weakly exact structures, a generalization of Quillen exact structures. We introduce weak counterparts of one-sided exact structures and show that a left and a right weakly exact structure generate a weakly exact structure. We further define weakly extriangulated structures on an additive category $\mathcal{A}$ and characterize weakly exact structures among them. We investigate when these structures on $\mathcal{A}$ form lattices. We prove that the lattice of substructures of a weakly extriangulated structure is isomorphic to the lattice of topologizing subcategories of a certain functor category. In the idempotent complete case, we characterize the lattice of all weakly exact structures and we prove the existence of a unique maximal weakly exact structure. We study in detail the situation when $\mathcal{A}$ is additively finite, giving a module-theoretic characterization of closed sub-bifunctors of $\mbox{Ext}^1$ among all additive sub-bifunctors.

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arXiv2023

A structural view of maximal green sequences

We study the structure of the set of all maximal green sequences of a finite-dimensional algebra. There is a natural equivalence relation on this set, which we show can be interpreted in several different ways, underscoring its significance. There are three partial orders on the equivalence classes, analogous to the partial orders on silting complexes and generalising the higher Stasheff--Tamari orders on triangulations of three-dimensional cyclic polytopes. We conjecture that these partial orders are in fact equal, just as the orders in the silting case have the same Hasse diagram. This can be seen as a refined and more widely applicable version of the No-Gap Conjecture of Brüstle, Dupont, and Perotin. We prove our conjecture in the case of Nakayama algebras.

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arXiv2023

Hereditary extriangulated categories: Silting objects, mutation, negative extensions

In this article, we initiate the study of hereditary extriangulated categories. Many important categories arising in representation theory in connection with various theories of mutation are hereditary extriangulated. Special cases include homotopy categories of 2-term complexes with projective components, which are related to silting mutation, and cluster categories (with relevant relative extriangulated structures) where cluster tilting mutation take place. We prove that there is a theory of irreducible mutation for maximal rigid objects and subcategories in hereditary extriangulated categories of dominant dimension 1. Applied to the examples above, this recovers 2-term silting mutation in triangulated categories and cluster tilting mutation. By constructing suitable extriangulated categories, we also recover tau-tilting mutation for gentle algebras and flips for their non-kissing facets. Combined with results by Adachi-Tsukamoto and Pauksztello-Zvonareva, our mutation also provides mutation for intermediate co-t-structures. In spirit of our earlier work, we study negative extensions in hereditary extriangulated categories. We give sufficient conditions for the existence of universal balanced negative extensions. We explicitly compute certain, a priori non-universal, versions of negative extensions in hereditary categories constructed from triangulated categories with rigid subcategories. We discuss examples where these two constructions give the same delta-functors and where they disagree.

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arXiv2021

Positive and negative extensions in extriangulated categories

We initiate the study of derived functors in the setting of extriangulated categories. By using coends, we adapt Yoneda's theory of higher extensions to this framework. We show that, when there are enough projectives or enough injectives, thus defined extensions agree with the ones defined earlier via projective or injective resolutions. For categories with enough projective or enough injective morphisms, we prove that these are right derived functors of the $\operatorname{Hom}$-bifunctor in either argument. Since $\operatorname{Hom}$ is only half-exact in each argument, it is natural to expect "negative extensions", i.e. its left derived functors, to exist and not necessarily vanish. We define negative extensions with respect to the first and to the second argument and show that they give rise to universal $δ$-functors, when there are enough projective or injective morphisms, respectively. In general, they are not balanced. However, for topological extriangulated categories, the existence of a balanced version of negative extensions follows from combining the work of Klemenc on exact $\infty$-categories with results of the second and third authors. We discuss various criteria under which one has the balance of the above bifunctors on the functorial or on the numerical levels. This happens, in particular, in the cases of exact or triangulated categories, and also in the case of the category of $n$-term complexes with projective components over a finite-dimensional algebra. Given a connected sequence of functors on an extriangulated category $(\mathcal{C},\mathbb{E},\mathfrak{s})$, we determine the maximal relative extriangulated structure, with respect to which the sequence is a $δ$-functor. We also find several equivalent criteria for the existence of enough projective or injective morphisms in a given extriangulated category.

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arXiv2020

Subword complexes and 2-truncated cubes

For a Coxeter element $c$ of a finite Coxeter group, we consider a family of subword complexes parameterized by reduced expressions of the longest element. This family generalizes $c-$cluster complexes. We describe vertices of these complexes in terms of roots of the corresponding root system. We prove that dual polytopes of all such complexes are combinatorial 2-truncated cubes.

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arXiv2020

Exact structures and degeneration of Hall algebras

We study degenerations of the Hall algebras of exact categories induced by degree functions on the set of isomorphism classes of indecomposable objects. We prove that each such degeneration of the Hall algebra $\mathcal{H}(\mathcal{E})$ of an exact category $\mathcal{E}$ is the Hall algebra of a smaller exact structure $\mathcal{E}' < \mathcal{E}$ on the same additive category $\mathcal{A}.$ When $\mathcal{E}$ is admissible in the sense of Enomoto, for any $\mathcal{E}' < \mathcal{E}$ satisfying suitable finiteness conditions, we prove that $\mathcal{H}(\mathcal{E}')$ is a degeneration of $\mathcal{H}(\mathcal{E})$ of this kind. In the additively finite case, all such degree functions form a simplicial cone whose face lattice reflects properties of the lattice of exact structures. For the categories of representations of Dynkin quivers, we recover degenerations of the negative part of the corresponding quantum group, as well as the associated polyhedral structure studied by Fourier, Reineke and the first author. Along the way, we give minor improvements to certain results of Enomoto and Brüstle-Langford-Hassoun-Roy concerning the classification of exact structures on an additive category. We prove that for each idempotent complete additive category $\mathcal{A}$, there exists an abelian category whose lattice of Serre subcategories is isomorphic to the lattice of exact structures on $\mathcal{A}$. We show that every Krull-Schmidt category admits a unique maximal admissible exact structure and that the lattice of smaller exact structures of an admissible exact structure is Boolean.

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