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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
Background: Multi-visceral resection often is used in the treatment of retroperitoneal sarcoma (RPS). The morbidity after distal pancreatectomy for primary pancreatic cancer is well-documented, but the outcomes after distal pancreatectomy for primary RPS are not. This study aimed to evaluate morbidity and oncologic outcomes after distal pancreatectomy for primary RPS. Methods: In this study, 26 sa
This Applied Optics feature issue on laser applications to chemical, security, and environmental analysis (LACSEA) highlights papers presented at the LACSEA 2020 Seventeenth Topical Meeting sponsored by The Optical Society (OSA).
For low-order adaptive optics systems, a method that is able to correct for system aberrations in the final focal plane is presented. The paper presents a novel figure of merit, corresponding to the integrated non-normalized tip-tilt-free optical transfer function. The inherent singular value decomposition modal content of the interaction matrix is used to optimize this figure of merit. The method
X-ray microscopy offers the opportunity to image biological and radiosensitive materials without special sample preparations, bridging optical and electron microscopy capabilities. However, the performance of such microscopes, when imaging radiosensitive samples, is not limited by their intrinsic resolution, but by the radiation damage induced on such samples. Here, we demonstrate a novel, to the
Gambling Disorder (GD) has recently been reclassified from an impulse-control disorder to a behavioural addiction and, as in other addictive disorders, the dopaminergic reward system is involved. According to neuroimaging studies, alterations within the striatal dopaminergic signalling can occur in GD. However, the findings to date are controversial and there has been no agreement yet on how the r
Background Utilizing surgical services, including caesarean sections, can result in catastrophic expenditure and impoverishment. In 2010, Sierra Leone introduced the Free Health Care Initiative (FHCI), a national financial risk protection program for the most vulnerable groups. Aim of this study was to investigate catastrophic expenditure and impoverishment related to caesarean section in Sierra L
