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Various faces of representation theory

Published 2000-01-21

In 2003 Dr Volodymyr Mazorchuk, Uppsala University received a STINT Institutional Grant for cooperation with Kyiv Taras Shevchenko University in Ukraine. The annual grant of SEK 150 000 is intended for cooperation in ‘Various faces of representation theory’.

Representation theory is the part of modern mathematics, which studies how a given mathematical object can be realized in terms of other, usually very well-understood, mathematical objects. For example, how a given ring (for example polynomials) can be realized in terms of linear operators on some vector space. The study of representations in terms of linear operators is perhaps the largest branch of the modern representation theory. It is also naturally divided into two parts: the first one, usually referred to as algebraic, studies usual linear operators on usual vector spaces; the second one, usually referred to as analytic, studies continuous linear operators on vector spaces with certain topology.

The project "Representation theory of algebras and applications", presently supported by STINT, unites the efforts of four research groups from

Department of Mathematics, Uppsala University;

Department of Mathematics, Chalmers University of Technology; Department of Mathematics, Kyiv University, Ukraine; Institute of Mathematics of the Ukrainian Academy of Sciences.

The members of these research groups represent both algebraic and analytic directions in representation theory, and the main idea of the project is to unite the strength of these groups to challenge new and old problems, especially those which are related to the common part of their research fields.

The principal problem in the representation theory is the following: given some object, a ring or an algebra say, describe all its possible realizations in terms of linear operators one works with. For roughly 25 years ago the algebraists have divided a very big class of algebras into two subclasses: tame and wild algebras. For tame algebras the above problem has a very precise solution, and for wild algebras its solution is roughly equivalent to the solution of this problems for all finitely generated algebras. For many years the specialists in analytic representation theory try to establish the same kind of dichotomy for analytic representations, but whether this is possible or not is still a very big puzzle, and one of the main questions the partners of the present project are trying to study.

The main activities of the project are visits of the involved researchers to each other, organization of mini-courses and also small workshops in the area. The principal idea behind these activities can be described as follows. Nowadays it is very usual that the research is strictly specialized. In many cases a researcher does not have enough time/will/competence to study and use the results, techniques and ideas, developed by other researchers even in rather close areas, as for example, algebraic and analytic branches of representation theory. To work against such tendencies one could try to organize interdisciplinary seminars, courses or conferences. The latter level is too general and usually does not really give a visible result.

In the present project we tried to put together four research groups, working originally in different directions of the modern representation theory. The possibility of visits and organization of mini-courses during these visits allows all the parties involved to intensively learn from each other and to have the up-to-date information about the adjoint research area. During the discussions new questions are posed and new problems are formulated. This intensifies the research and makes it cover a wider area, which all the members of our research groups experienced during the first year of the project. The results of the executed research can then be presented at the mini-workshops, which are also organized within the frame of the project. On this stage there is also a possibility to invite some specialists from outside the original research groups, which is very useful for presenting own results, learning about the results obtained by other people, and for sharing the ideas in general. In December 2003 the project allowed the partners to organize a workshop in Kyiv, and in November 2004 we were organizing the second "Tame and Wild Workshop" in Uppsala, where we, in particular, discussed and reported the results obtained during the first year of the project.

Volodymyr Mazorchuk
Department of Mathematics
Uppsala University

Senast uppdaterad: 05-07-15 09:55

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