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Erfarenheterna av att arbeta digitalt är nu många och användbara. Pandemin slog hårt, men den har också lett till att digitala barnsjukdomar kunnat utrotas, skriver de två forskarna Claire Hoolohan och Tullia Jack.
Inspired by work of Ladkani and Groth–Šťovíček, we explain how to construct generalisations of the classical reflection functors of Bernšteĭn, Gel′fand, and Ponomarev by means of the Grothendieck construction.
We show that the perfect derived categories of Iyama’s d-dimensional Auslander algebras of type A are equivalent to the partially wrapped Fukaya categories of the d-fold symmetric product of the 2-dimensional unit disk with finitely many stops on its boundary. Furthermore, we observe that Koszul duality provides an equivalence between the partially wrapped Fukaya categories associated to the d-fol
These notes are an expanded version of my talk at the ICRA 2018 in Prague, Czech Republic; they are based on joint work with Tobias Dyckerhoff and Tashi Walde. In them we relate Iyama's higher Auslander algebras of type A to Eilenberg-Mac Lane spaces in algebraic topology and to higher-dimensional versions of the Waldhausen S-construction from algebraic K-theory.
We develop a novel combinatorial perspective on the higher Auslander algebras of type A, a family of algebras arising in the context of Iyama's higher Auslander–Reiten theory. This approach reveals interesting simplicial structures hidden within the representation theory of these algebras and establishes direct connections to Eilenberg–MacLane spaces and higher-dimensional versions of Waldhausen's
We introduce higher dimensional analogues of the Nakayama algebras from the viewpoint of Iyama's higher Auslander–Reiten theory. More precisely, for each Nakayama algebra A and each positive integer d, we construct a finite dimensional algebra A(d) having a distinguished d-cluster-tilting -module whose endomorphism algebra is a higher dimensional analogue of the Auslander algebra of A. We also con
This article consists of an introduction to Iyama's higher Auslander–Reiten theory for Artin algebras from the viewpoint of higher homological algebra. We provide alternative proofs of the basic results in higher Auslander–Reiten theory, including the existence of d-almost-split sequences in d-cluster-tilting subcategories, following the approach to classical Auslander–Reiten theory due to Ausland
The class of support τ-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article, we study τ-tilting finite algebras, that is, finite dimensional algebras A with finitely many isomorphism classes of indecomposable τ-rigid modules. We show that A is τ-tilting finite if and only if every torsion class in modA is functo
Let R be a commutative artinian ring. We extend higher Auslander correspondence from Artin R-algebras of finite representation type to dualizing R-varieties. More precisely, for a positive integer d, we show that a dualizing R-variety is d-abelian if and only if it is a d-Auslander dualizing R-variety if and only if it is equivalent to a d-cluster-tilting subcategory of the category of finitely pr
We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories sat
We show that the category of finitely generated free modules over certain local rings is 𝑛-angulated for every 𝑛⩾3. In fact, we construct several classes of 𝑛-angles, parameterized by equivalence classes of units in the local rings. Finally, we show that for odd values of 𝑛 some of these 𝑛-angulated categories are not algebraic.
The class of support τ -tilting modules was introduced recently by Adachi et al. These modules complete the class of tilting modules from the point of view of mutations. Given a finite-dimensional algebra A, we study all basic support τ -tilting A-modules which have a given basic τ -rigid A-module as a direct summand. We show that there exist an algebra C such that there exists an order-preserving
Let X be a weighted projective line and coh X the associated category of coherent sheaves. We classify the tilting complexes T in Db(coh X) such that τ2T ≅ T , where τ is the Auslander–Reiten translation in Db(coh X). As an application of this result, we classify the 2-representation-finite algebras which are derived-equivalent to a canonical algebra. This complements Iyama-Oppermann’s classificat
Given a connected non-negative unit form we construct an extended affine Lie algebra by giving a Chevalley basis for it. We also obtain this algebra as a quotient of an algebra defined by means of generalized Serre relations by M. Barot, D. Kussin and H. Lenzing. This is done in an analogous way to the construction of the simply-laced affine Kac–Moody algebras. Thus, we obtain a family of extended
Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the automorphism group of the corresponding cluster category and second by giving explicit sequences of mutations.
This paper surveys recent contructions in higher Auslander–Reiten theory. We focus on those which, due to their combinatorial properties, can be regarded as higher dimensional analogues of path algebras of linearly oriented type A quivers. These include higher dimensional analogues of Nakayama algebras, of the mesh category of type ZA∞ and the tubes, and of the triangulated category generated by a
