arXiv subjects

George Dimitrov

Publications and source records attributed to George Dimitrov.

10 recordsLinked to original sources

arXiv2022

Non-commutative counting and stability

The second author and Katzarkov introduced categorical invariants based on counting of full triangulated subcategories in a given triangulated category $\mathcal T$, and they demonstrated different choices of additional properties of the subcategories being counted, in particular - an approach to make non-commutative counting in $\mathcal T$ dependable on a stability condition $σ\in {\rm Stab}(\mathcal T)$. In this paper, we focus on this approach. After recalling the definitions of a stable non-commutative curve in $\mathcal T$ and related notions, we prove a few general facts and study an example: $\mathcal T = D^b(Q)$, where $Q$ is the acyclic triangular quiver. In previous papers, it was shown that there are two non-commutative curves of non-commutative genus $1$ and infinitely many non-commutative curves of non-commutative genus $0$ in $D^b(Q)$. Our studies here imply that for an open and dense subset in ${\rm Stab}(D^b(Q))$ the stable non-commutative curves in $D^b(Q)$ are finitely many. This paper also introduces counting of semistable derived points and shows that the corresponding invariants are finite on an open dense subset of ${\rm Stab}\big(D^b(Q)\big)$.

math.CT↗

arXiv2020

Non-commutative counting invariants and curve complexes

In our previous paper, viewing $D^b(K(l))$ as a non-commutative curve, where $K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical invariants via counting of non-commutative curves. Roughly, these invariants are sets of subcategories in a given category and their quotients. The non-commutative curve-counting invariants are obtained by restricting the subcategories to be equivalent to $D^b(K(l))$. The general definition defines much larger class of invariants and many of them behave properly with respect to fully faithful functors. Here, after recalling the definition, we focus on examples and extend our studies beyond counting. We enrich our invariants with structures: the inclusion of subcategories makes them partially ordered sets, and considering semi-orthogonal pairs of subcategories as edges amount to directed graphs. In addition to computing the non-commutative curve-counting invariants in $D^b(Q)$ for two affine quivers, for $ A_n$ and $D_4$ we derive formulas for counting of the subcategories of type $D^b(A_k)$ in $D^b(A_n)$, whereas for the two affine quivers and for $D_4$ we determine and count all generated by an exceptional collection subcategories. Estimating the numbers counting non-commutative curves in $D^b({\mathbb P}^2)$ modulo group action we prove finiteness and that an exact determining of these numbers leads to proving (or disproving) of Markov conjecture. Regarding the mentioned structure of a partially ordered set we initiate intersection theory of non-commutative curves. Via the structure of a directed graph we build an analogue to the classical curve complex used in Teichmueller and Thurston theory. The paper contains many pictures of graphs and presents an approach to Markov Conjecture via counting of subgraphs in a graph associated with $D^b(P^2)$. Some of the results proved here were announced in the previous work.

math.CT↗

arXiv2019

More finite sets coming from non-commutative counting

In our previous papers we introduced categorical invariants, which are, roughly speaking, sets of triangulated subcategories in a given triangulated category and their quotients. Here is extended the list of examples, where these sets are finite. Using results by Geigle, Lenzning, Meltzer, Hübner for weighted projective lines we show that for any two affine acyclic quivers $Q$, $Q'$ (i.e. quivers of extended Dynkin type) there are only finitely many full triangulated subctegories in $D^b(Rep_{\mathbb K}(Q))$, which are equivalent to $D^b(Rep_{\mathbb K}(Q'))$, where ${\mathbb K}$ is an algebraically closed field. Some of the numbers counting the elements in these finite sets are explicitly determined.

math.CT↗

arXiv2018

Non-semistable exceptional objects in hereditary categories: some remarks and conjectures

In our previous paper we studied non-semistable exceptional objects in hereditary categories and introduced the notion of regularity preserving category, but we obtained quite a few examples of such categories. Certain conditions on the Ext-nontrivial couples (exceptional objects $X,Y\in \mathcal A$ with ${\rm Ext}^1(X,Y)\neq 0$ and ${\rm Ext}^1(Y,X)\neq 0$) were shown to imply regularity-preserving. This paper is a brief review of the previous paper (with emphasis on regularity preserving property) and we add some remarks and conjectures. It is known that in Dynkin quivers ${\rm Hom}(ρ,ρ')=0$ or ${\rm Ext}^1(ρ,ρ')=0$ for any two exceptional representations. In the present paper we use this property to show that for any Dynkin quiver $Q$ there are no Ext-nontrivial couples in $Rep_k(Q)$, which implies regularity preserving of $Rep_k(Q)$, where $k$ is an algebraically closed field. We study this property in other quivers. In particular in any star quiver with three arms $Q$ for any two exceptional representations $ρ, ρ'$ we have ${\rm Hom}(ρ,ρ')=0$ or ${\rm Ext}^1(ρ,ρ')=0$ provided that $ρ$ or $ρ'$ is a thin representation. In the previous version we asserted falsely that this holds for any two exceptional representations (without imposing the restriction that one of them is thin) for extended Dynkin quivers $\widetilde{\mathbb E}_6, \widetilde{\mathbb E}_7, \widetilde{\mathbb E}_8 $.

math.CT↗

arXiv2017

Some new categorical invariants

We introduce several notions and give examples. We prove that ${\rm Stab}(D^b(K(l)))\cong {\mathbb C}\times \mathcal H$ for $l\geq 3$, where $K(l)$ is $l$-Kronecker quiver. This is an example of SOD, where ${\rm Stab}( \langle \mathcal T_1,\mathcal T_2\rangle )\not \cong{\rm Stab}(\mathcal T_1)\times {\rm Stab}(\mathcal T_2)$. This example suggest a new notion of a norm, strictly increasing on $\{D^b(K(l))\}_{l\geq 2}$. To a triangulated category $\mathcal T$ which has property of a phase gap we attach a non-negative number $\Vert \mathcal T \Vert_{\varepsilon}$. Natural assumptions on a SOD imply $ \Vert \langle \mathcal T_1,\mathcal T_2\rangle \Vert_{\varepsilon}\geq {\rm max}\{ \Vert \mathcal T_1 \Vert_{\varepsilon}, \Vert\mathcal T_2 \Vert_{\varepsilon}\}$. Using this we define a topology on the set of equivalence classes of triangulated categories with a phase gap, where the set of discrete derived categories is a discrete subset and the rationality of a smooth surface $S$ ensures that $[D^b(point)] \in {\rm Cl}([D^b(S)])$. Viewing $D^b(K(l))$ as a non-commutative curve, we observe that it is reasonable to count non-commutative curves in any category in a small neighborhood of $D^b(K(l))$. Examples show that this idea (non-commutative curve-counting) opens directions to new categorical structures and connections to number theory and classical geometry. We give a definition, which specializes to the non-commutative curve-counting invariants. In an example arising on the A side we specialize our definition to non-commutative Calabi-Yau curve-counting, where the entities we count are a Calabi-Yau modification of $D^b(K(l))$. Finally we speculate that one might consider a holomorphic family of categories, introduced by Kontsevich, as a non-commutative extension with the norm playing a role similar to the classical notion of degree of an extension in Galois theory.

math.CT↗

arXiv2014

Bridgeland stability conditions on the acyclic triangular quiver

Using results in a previous paper "Non-semistable exceptional objects in hereditary categories", we focus here on studying the topology of the space of Bridgeland stability conditions on $D^b(Rep_k(Q ))$, where $Q$ is the acyclic triangular quiver (the underlying graph is the extended Dynkin diagram $\widetilde{\mathbb A}_2$). In particular, we prove that this space is contractible (in the previous paper it was shown that it is connected).

math.CT↗

arXiv2013

Non-semistable exceptional objects in hereditary categories

For a given stability condition $σ$ on a triangulated category we define a $σ$-exceptional collection as an Ext-exceptional collection, whose elements are $σ$-semistable with phases contained in an open interval of length one. If there exists a full $σ$-exceptional collection, then $σ$ is generated by this collection in a procedure described by E. Macrì. Constructing $σ$-exceptional collections of length at least three in $D^b(\mathcal A)$ from a non-semistable exceptional object, where $\mathcal A$ is a hereditary hom-finite abelian category, we introduce certain conditions on the Ext-nontrivial couples (couples of exceptional objects $X,Y\in \mathcal A$ with ${\rm Ext}^1(X,Y)\neq 0$ and ${\rm Ext}^1(Y,X)\neq 0$). After a detailed study of the exceptional objects of two tame quivers $Q_1$ and $Q_2$ with three and four vertices, respectively, we observe that the needed conditions do hold in $Rep_k(Q_1)$, $Rep_k(Q_2)$, where $k$ is an algebraically closed field. Combining these findings, we prove that for each $σ\in {\rm Stab}(D^b(Q_1))$ there exists a full $σ$-exceptional collection. It follows that ${\rm Stab}(D^b(Q_1))$ is connected.

math.CT↗

arXiv2013

Dynamical systems and categories

We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A-infinity categories. First, entropy is defined for exact endofunctors and computed in a variety of examples. In particular, the classical entropy of a pseudo-Anosov map is recovered from the induced functor on the Fukaya category. Second, the density of the set of phases of a Bridgeland stability condition is studied and a complete answer is given in the case of bounded derived categories of quivers. Certain exceptional pairs in triangulated categories, which we call Kronecker pairs, are used to construct stability conditions with density of phases. Some open questions and further directions are outlined as well.

math.CT↗

arXiv2012

Homogeneous Hypercomplex Structures I - the compact Lie groups

We introduce a remarkable subset "the stem" of the set of positive roots of a reduced root system. The stem determines several interesting decompositions of the corresponding reductive Lie algebra. It gives also a nice simple three dimensional subalgebra and a "Cayley transform". In the present paper we apply the above devices to give a complete classification of invariant hypercomplex structures on compact Lie groups.

math.DG↗

arXiv2012

Homogeneous Hypercomplex Structures II - Coset Spaces of compact Lie Groups

We obtain a complete classification of hypercomplex manifolds, on which a compact group of automorphisms acts transitively. The description of the spaces as well as the proofs of our results use only the structure theory of reductive groups, in particular the notion of "stem" of a reduced root system, introduced in the first paper of this series.

math.DG↗