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This Research in Groups aims at master students in the programs International Master in Mathematics, in Mathematics, and Mathematical Physics. It may also be interesting for students in the master program Physics.

In mathematical models of physical systems, the physical observables
are typically described by C^{*}-algebras or von Neumann
algebras. While this gives a very elegant and powerful spectral
calculus, many situations will not directly yield a such nice classes
of algebras. In various quantization theories the construction of
C^{*}-algebras is difficult or unclear. One way out is to focus on the
algebraic features before taking into account the analytic issues as
well. This is the main motivation for considering ^{*}-algebras
without any analysis involved. Beyond quantization theories, other
important examples are group algebras or universal enveloping algebras
of Lie algebras but also algebras of differential operators. Here one
typically has by no means a C^{*}-norm available.

The aim of this RiG is to find a common algebraic framework for a
reasonable representation theory of such algebras. It turns out that
aspects of positivity can be formulated in a entirely algebraic way
yielding interesting structures for the representation theory. In
general, it will be difficult if not impossible to understand the
representation theory of a given algebra completely. However, and this
is quite surprising, it might be possible to compare it to the
representation theory of a different algebra and determine whether or
not the two algebras have the same representation theory. This is the
main task of Morita theory, which we will present both in a purely
ring-theoretic context and in an adapted version for ^{*}-algebras.
We will develop the necessary category-theoretic notions to put the
question of Morita equivalence in the right perspective.

The course consists of essentially two components: first, a lecture where the basic notions of representation theory are explained. The preliminary program for the lecture component includes the following topics:

^{*}-Algebras- Elementary representation theory and GNS construction
- Pre Hilbert modules
- K
_{0}-Theory in several versions - Internal and external tensor products
- Equivalence bimodules and Morita theory in various flavours
- Bicategorical approach
- Picard bigroupoid
- Morita invariants
- Deformation theory, Hermitean and completely positive deformations
- Deformation of modules and inner products
- Stability of K
_{0}under formal deformations - Classical limit of representation theories and equivalence bimodules

The second component will be a seminar by the students on more particular topics. We expect the participants to write a small proceeding-like summary of their seminar talks.

For this RiG you will not need much prerequisites. In fact, a good knowledge in (multi-) linear algebra will be sufficient for most things. Depending on the examples you want to discuss, some ideas about Lie algebras, differential geometry, algebra, functional analysis might be useful but certainly not necessary.

However, if you are in doubt, please contact me directly and we will find a solution. I will explain the necessary things either in the lecture part or directly.

The following list of references will be discussed in the first meeting. More references will be given individually.

If there are collisions with other lectures or seminars, please contact me early: maybe one can still shift things around a bit.

- Lecture: Friday 10 --12, SE 40.
- Seminar: The seminar will take place at one or two days at the end of the semester. The precise date will be announced.

- On WueCampus there will be a homepage for this RiG. The WueCampus homepage will replace this site as soon as it is activated.

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