24,048 research outputs found
A generic algebra associated to certain Hecke algebras
We initiate the systematic study of endomorphism algebras of permutation
modules and show they are obtainable by a descent from a certain "generic"
Hecke algebra, infinite-dimensional in general, coming from the universal
enveloping algebra of gl_n (or sl_n). The endomorphism algebras and the generic
algebras are cellular. We give several equivalent descriptions of these
algebras, find a number of explicit bases, and describe indexing sets for their
irreducible representations.Comment: 19 page
Silted algebras
We study endomorphism algebras of 2-term silting complexes in derived
categories of hereditary finite dimensional algebras, or more generally of
-finite hereditary abelian categories. Module
categories of such endomorphism algebras are known to occur as hearts of
certain bounded -structures in such derived categories. We show that the
algebras occurring are exactly the algebras of small homological dimension,
which are algebras characterized by the property that each indecomposable
module either has injective dimension at most one, or it has projective
dimension at most one.Comment: Fix some typos, to appear in Adv. Mat
Hodge structures of type (n,0,...,0,n)
This paper determines all the possible endomorphism algebras for polarizable
Q-Hodge structures of type (n,0,...,0,n). This generalizes the classification
of the possible endomorphism algebras of abelian varieties by Albert and
Shimura. As with abelian varieties, the most interesting feature of the
classification is that in certain cases, every Hodge structure on which a given
algebra acts must have extra endomorphisms.Comment: 19 page
On derived equivalences of lines, rectangles and triangles
We present a method to construct new tilting complexes from existing ones
using tensor products, generalizing a result of Rickard. The endomorphism rings
of these complexes are generalized matrix rings that are "componentwise" tensor
products, allowing us to obtain many derived equivalences that have not been
observed by using previous techniques.
Particular examples include algebras generalizing the ADE-chain related to
singularity theory, incidence algebras of posets and certain Auslander algebras
or more generally endomorphism algebras of initial preprojective modules over
path algebras of quivers. Many of these algebras are fractionally Calabi-Yau
and we explicitly compute their CY dimensions. Among the quivers of these
algebras one can find shapes of lines, rectangles and triangles.Comment: v3: 21 pages. Slight revision, to appear in the Journal of the London
Mathematical Society; v2: 20 pages. Minor changes, pictures and references
adde
Derived autoequivalences from periodic algebras
We present a construction of autoequivalences of derived categories of
symmetric algebras based on projective modules with periodic endomorphism
algebras. This construction generalises autoequivalences previously constructed
by Rouquier-Zimmermann and is related to the autoequivalences of Seidel-Thomas
and Huybrechts-Thomas. We show that compositions and inverses of these
equivalences are controlled by the resolutions of our endomorphism algebra and
that each autoequivalence can be obtained by certain compositions of derived
equivalences between algebras which are in general not Morita equivalent.Comment: 34 pages; v2 is post referee report. The biggest changes from v1 are
in Section 5.2. Final version has appeared in Proc. LM
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