08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access Gendo-Frobenius algebras and comultiplication(2022) Yırtıcı, ÇiğdemGendo-Frobenius algebras are a common generalisation of Frobenius algebras and of gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius and on gendo-symmetric algebras. In addition, Frobenius algebras are shown to be precisely those gendo-Frobenius algebras that have a counit compatible with this comultiplication. Moreover, a new characterisation of gendo-Frobenius algebras is given. This new characterisation is a key for constructing the comultiplication of gendo-Frobenius algebras.Item Open Access On quiver Grassmannians and orbit closures for gen-finite modules(2021) Pressland, Matthew; Sauter, JuliaWe show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.Item Open Access Dominant and global dimension of blocks of quantised Schur algebras(2021) Fang, Ming; Hu, Wei; Koenig, SteffenGroup algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and Sq(n,r) with n⩾r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59-85, 2021).Item Open Access Brauer graph algebras are closed under derived equivalence(2022) Antipov, Mikhail; Zvonareva, AlexandraIn this paper the class of Brauer graph algebras is proved to be closed under derived equivalence. For that we use the rank of the maximal torus of the identity component Out0(A)of the group of outer automorphisms of a symmetric stably biserial algebra A .Item Open Access Lifting and restricting t‐structures(2022) Marks, Frederik; Zvonareva, AlexandraWe explore the interplay between t-structures in the bounded derived category of finitely presented modules and the unbounded derived category of all modules over a coherent ring 𝐴 using homotopy colimits. More precisely, we show that every intermediate t-structure in 𝐷𝑏(mod(𝐴)) can be lifted to a compactly generated t-structure in 𝐷(Mod(𝐴)), by closing the aisle and the coaisle of the t-structure under directed homotopy colimits. Conversely, we provide necessary and sufficient conditions for a compactly generated t-structure in 𝐷(Mod(𝐴)) to restrict to an intermediate t-structure in 𝐷𝑏(mod(𝐴)), thus describing which t-structures can be obtained via lifting. We apply our results to the special case of HRS t-structures. Finally, we discuss various applications to silting theory in the context of finite dimensional algebras.Item Open Access A characterisation of Morita algebras in terms of covers(2021) Cruz, TiagoA pair (A, P) is called a cover of EndA(P)op if the Schur functor HomA(P,-) is fully faithful on the full subcategory of projective A-modules, for a given projective A-module P. By definition, Morita algebras are the covers of self-injective algebras and then P is a faithful projective-injective module. Conversely, we show that A is a Morita algebra and EndA(P)op is self-injective whenever (A, P) is a cover of EndA(P)op for a faithful projective-injective module P.Item Open Access Lattices of t‐structures and thick subcategories for discrete cluster categories(2023) Gratz, Sira; Zvonareva, AlexandraWe classify t-structures and thick subcategories in any discrete cluster category C(Z) of Dynkin type 𝐴, and show that the set of all t-structures on C(Z) is a lattice under inclusion of aisles, with meet given by their intersection.We show that both the lattice of t-structures on C(Z) obtained in this way and the lattice of thick subcategories of C(Z) are intimately related to the lattice of non-crossing partitions of type 𝐴. In particular, the lattice of equivalence classes of non-degenerate tstructures on such a category is isomorphic to the lattice of non-crossing partitions of a finite linearly ordered set.Item Open Access Higher Morita-Tachikawa correspondence(2024) Cruz, TiagoImportant correspondences in representation theory can be regarded as restrictions of the Morita–Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo‐symmetric algebras. Explicitly, the Morita-Tachikawa correspondence describes that endomorphism algebras of generators-cogenerators over finite‐dimensional algebras are exactly the finite‐dimensional algebras with dominant dimension at least two. In this paper, we introduce the concepts of quasi‐generators and quasi‐cogenerators that generalise generators and cogenerators, respectively. Using these new concepts, we present higher versions of the Morita-Tachikawa correspondence that take into account relative dominant dimension with respect to a self‐orthogonal module with arbitrary projective and injective dimensions. These new versions also hold over Noetherian algebras that are finitely generated and projective over a commutative Noetherian ring.Item Open Access Centers of Hecke algebras of complex reflection groups(2023) Chavli, Eirini; Pfeiffer, GötzWe provide a dual version of the Geck-Rouquier Theorem (Geck and Rouquier in Finite Reductive Groups (Luminy, 1994), Progr. Math., vol. 141, Birkhäuser Boston, Boston, pp. 251–272, 1997) on the center of an Iwahori-Hecke algebra, which also covers the complex case. For the eight complex reflection groups of rank 2, for which the symmetrising trace conjecture is known to be true, we provide a new faithful matrix model for their Hecke algebra H . These models enable concrete calculations inside H . For each of the eight groups, we compute an explicit integral basis of the center of H .