Triangulated categories of matrix factorizations for regular systems of weights with \(\varepsilon = - 1\)
type: Item
instance of: scholarly article
title: Triangulated categories of matrix factorizations for regular systems of weights with \(\varepsilon = - 1\)
zbMATH Open document ID: 1172.18002
author: Hiroshige Kajiura
author: Kyoji Saito
author: Atsushi Takahashi
published in: Advances in Mathematics
publication date: 2009-03-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/0708.0210
review text: One of the results of this article is a categorification of Arnold's strange duality for exceptional unimodular singularities; cf.\ Cor.\ 5.12. Let \(W = (a,b,c;h)\), where \(a,b,c,h\in\mathbb Z_{\geq 1}\) such that \(\gcd(a,b,c)=1\) and \(a+b+c-h= -1\). Suppose that \(T^{-h}(T^h-T^a) (T^h-T^b) (T^h-T^c) (T^a-1)^{-1} (T^b-1)^{-1} (T^c-1)^{-1}\) has no poles except at \(T=0\), and that its Laurent series at \(T = 0\) has coefficient \(0\) for \(T^0\). Consider \(A = \mathbb C[x,y,z]\). Let \(E = h^{-1}(ax{\text{D}}_x + by{\text{D}}_y + cz{\text{D}}_z)\). Let \(A_s\) denote the eigenspace of \(E\) in \(A\) of eigenvalue \(s/2\). We obtain the grading \(A = \bigoplus_{k\in\mathbb Z_{\geq 0}} A_{2k/h}\). Let \(f_W\in A_2\) be such that \(\dim_{\mathbb C} A/({\text{D}}_x f_W,{\text{D}}_y f_W,{\text{D}}_z f_W)\) is finite. If the (combinatorially defined) signature of \(W\) has length 3, then \(f_W\) has an exceptional unimodular singularity at \((x,y,z)=(0,0,0)\). Define the Frobenius category \(\text{MF}^{\text{gr}}_A(f_W)\) (matrix factorisations) to have as objects diagrams \(F_0\rightarrow{f_0} F_1\rightarrow{f_1} F_0\), where \(F_0\) and \(F_1\) are f.g.\ free graded \(A\)-modules, \(f_0\) is a graded \(A\)-linear map of degree \(0\), \(f_1\) of degree \(2\), and where \(f_1\circ f_0 = f_W\cdot{\text{id}}\) and \(f_0\circ f_1 = f_W\cdot{\text{id}}\). Morphisms are compatible pairs of morphisms at \(F_0\) and \(F_1\); admissible short exact sequences are defined pointwise. Write \({\mathcal T}_W\) for the stable category of \(\text{MF}^{\text{gr}}_A(f_W)\). The main result of this article is the construction of a full strongly exceptional sequence for \({\mathcal T}_W\), and, consequently, of an exact equivalence of \({\mathcal T}_W\) with the bounded derived category of an algebra explicitly described via quiver and relations. The authors state that this parallels a result of \textit{H. Lenzing} and \textit{J. de la Pena} [``Extended canonical algebras and Fuchsian …
reviewed by: Matthias Künzer
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 5530601
zbMATH Keywords: singularity
zbMATH Keywords: mirror symmetry
zbMATH Keywords: strange duality
zbMATH Keywords: representations of finite-dimensional algebras
zbMATH Keywords: triangulated category
MaRDI profile type: Publication
OpenAlex ID: W2962894525
arXiv ID: 0708.0210
Wikidata QID: Q114211696
cites work: CRITICAL POINTS OF SMOOTH FUNCTIONS AND THEIR NORMAL FORMS
cites work: Representation theory of artin algebras VI: A functorial approach to aimost split sequences
cites work: Q3769015
cites work: Q3972472
cites work: ENHANCED TRIANGULATED CATEGORIES
cites work: HOMOLOGICAL PROPERTIES OF ASSOCIATIVE ALGEBRAS: THE METHOD OF HELICES
cites work: Stability conditions on triangulated categories
cites work: Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity
cites work: Quotient-conical singularities on complex surfaces
cites work: An arithmetic characterisation of the symmetric monodromy groups of singularities
cites work: Q3712441
cites work: Homological Algebra on a Complete Intersection, with an Application to Group Representations
cites work: Q3889229
cites work: Intersection matrices for certain singularities
cites work: Dynkin diagrams for unimodal singularities
cites work: Q3795770
cites work: Perpendicular categories with applications to representations and sheaves
cites work: Q3994341
cites work: Matrix factorizations and representations of quivers. II: Type \(ADE\) case.
cites work: Q4335023
cites work: Homological Algebra of Mirror Symmetry
cites work: Extended canonical algebras and Fuchsian singularities.
cites work: Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities
cites work: Extended affine root system. IV: Simply-laced elliptic Lie algebras
cites work: Q3029121
cites work: Duality for regular systems of weights
cites work: Q3765947
cites work: Q2762153
cites work: K. Saito's duality for regular weight systems and duality for orbifoldized Poincaré polynomials
cites work: Weighted Projective Lines Associated to Regular Systems of Weights of Dual Type
cites work: Q3995215
DOI: 10.1016/J.AIM.2008.11.001
description: scientific article