General theory of algebras

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A general approach to finite-dimensional division algebras

Get permission to re-use this article. Create citation alert. Buy this article in print. The determination of conjugacy classes within the group of all automorphisms of an affine Kac-Moody algebra is discussed in detail, and the limited role of Cartan preserving automorphisms is emphasized.

ISBN 13: 9780444507518

A comprehensive method of dealing with all the automorphisms of an untwisted affine Kac-Moody algebra is developed. This is based on a matrix formulation of the untwisted affine Kac-Moody algebras. As applications, we establish the relative version of Brenner-Butler's tilting theorem and higher Auslander correspondence. Examples will be given to explain the main results.

This is joint work with Julia Sauter. This will include connections to the general theory of trace modules with some preliminary applications to ring classifications and conjectures involving modules with no self-extensions. Using a recent result of Geuenich, we can describe all tilting modules as universal extensions of full exceptional sequences. Then we use spherical twists to construct also tilting complexes in the derived category, which have a very explicit description.

It is still open in general whether this is already the full classification, so we discuss the known results and the open problems. This is joint work with David Ploog. By contrast, the Picard group of a finite group algebra over a p-adic ring with finite residue field is a finite group. The structure of the automorphism group of a finite group algebra over a p-local domain with an algebraically closed residue field is largely unknown; it seems to be unknown in general whether this group is finite.

A recent result by F.

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Eisele shows that this group is also an algebraic group over the residue field. In joint work with R. Boltje and R. This is partly motivated by the - to date open - question whether Morita equivalent block algebras have isomorphic defect groups and fusion systems.

General Theory of C*-Algebras: Vol. 3 by Corneliu Constantinescu -

Another motivation comes from recent work of Eaton, Eisele, and Livesey, where the above results on Picard groups play a role in the proof of special cases of Donovan's finiteness conjecture. A theorem of Deligne says that in characteristic zero, any symmetric tensor category "of moderate growth" admits a tensor functor to vector spaces or to super i. In prime characteristic, this is not true, but one may ask whether there is a good list of "incompressible" symmetric tensor categories to which they they do all map.

We construct an infinite ascending chain of finite symmetric tensor categories in characteristic two, all of which are incompressible. Similar examples should exist in other prime characteristics, but the details have not yet been worked out. The object is to establish a Green correspondence for categories of complexes of modules as well as their homotopy categories and derived categories.

There is a categorical expression of the Green correspondence that is similar to a construction of Benson and Wheeler. At a key point in the constuction, we must assume that one of the categories has idempotent completion. This condition holds provided the category has countable direct summands.

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But under that assumption there are many categories that satisfy the hypothesis. Wednesday, 06 February , Room U Teresa Conde Stuttgart : Gabriel-Roiter measure and finiteness of the representation dimension Abstract: The induction scheme used in Roiter's proof of the first Brauer-Thrall conjecture prompted Gabriel to introduce an invariant, known as the Gabriel—Roiter measure. The usefulness of the Gabriel-Roiter measure is not limited to the first Brauer-Thrall conjecture: Ringel has used it to give new proofs of results established by Auslander in the 70's.

In this talk, we use the Gabriel-Roiter measure to provide a new proof of the finiteness of the representation dimension for Artin algebras, a result originally proved by Iyama in Friday, 01 February , Room U Severin Barmeier Bonn : Diagrams of algebras, categories of coherent sheaves and deformations Abstract: Given a complex algebraic variety X, the restriction of its structure sheaf to a finite cover of affine open sets can be viewed as a diagram of commutative algebras.

Deformations of a diagram obtained in this way correspond precisely to deformations of the category of quasi coherent sheaves as an Abelian category after W. Lowen and M. Van den Bergh. We describe the higher deformation theory explicitly via L-infinity algebras for X covered by two affine opens and explain the connection to "classical" deformations of the complex structure and deformation quantizations by means of examples. This is joint work with Y. My talk is concerned with certain 'non-commutative curve singularities' arising from ribbon graphs on closed surfaces.

Universal algebra

These so-called ribbon graph orders can be viewed as infinite-dimensional versions of Brauer graph algebras as well as gentle algebras. Although defined by particular combinatorial conditions, it turns out that ribbon graph orders possess a unique blend of purely homological features such as a Calabi-Yau property of the perfect derived category and semisimplicity of the singularity category.

Using their homological characterization I will show that ribbon graph orders as well as certain finite-dimensional special biserial algebras are preserved under derived equivalences.

Friday, 25 January , Room U Sebastian Opper Paderborn : On auto-equivalences and derived invariants of gentle algebras Abstract: This talk will be about derived equivalences of gentle algebras and the group of auto-equivalences of their derived categories. In joint work with Plamondon and Schroll, we attached a surface to every gentle algebra and showed that its geometry encodes of the triangulated structure of the derived category.

This is opposed to geometric theory which involves partial functions or binary relationships or existential quantors - see e. Euclidean geometry where the existence of points or lines is postulated. An Algebraic Theory T is a category whose objects are natural numbers 0, 1, 2, This allows interpreting n as a cartesian product of n copies of 1. Let's define an algebraic theory T taking hom n , m to be m -tuples of polynomials of n free variables X 1 , In this case proj i is the same as X i.

This theory T is called the theory of commutative rings.

General theory of algebras General theory of algebras
General theory of algebras General theory of algebras
General theory of algebras General theory of algebras
General theory of algebras General theory of algebras
General theory of algebras General theory of algebras
General theory of algebras General theory of algebras
General theory of algebras General theory of algebras
General theory of algebras General theory of algebras

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