In Algebraic Quantum Field Theory (AQFT) one associates to regions O in a spacetime manifold von Neumann algebras corresponding to the observables measurable in the laboratory O (local observables). For free theories, this data can be obtained by Second Quantization from nets of standard subspaces which are naturally associated to anti-unitary representations of Lie groups (the one-particle representations). We shall discuss how non-trivial examples of such nets can be constructed on causal symmetric spaces. Here a key point is the concept of a ``wedge domain'' in a symmetric space and how these domains can be derived from Lie algebraic data, such as three gradings, involutions on euclidean Jordan algebras etc.

This is joint work with Gestur Olafsson (Baton Rouge) and Vincenzo Morinelli (Erlangen).

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15:00 - 15:55:   Effie PAPAGEORGIOU (Universty of Crete)
Title: Asymptotic behavior for the heat equation on noncompact symmetric spaces

We study the long-time asymptotic behaviour of solutions to the heat equation, on noncompact symmetric spaces G/K. When the initial data is integrable and K-bi-invariant, we prove via spherical analysis that the solution converges in L¹ norm to the heat kernel, times the mass of the initial data. However, for general integrable solutions to the heat equation, the result may fail.

Work in progress, in collaboration with J.-Ph. Anker (Orléans), H.-W. Zhang (Ghent).

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16:15 - 17:10:   Dražen ADAMOVIĆ (University of Zagreb)
Title: On vertex-algebraic proof of complete reducibility of certain categories of modules for affine Lie algebras

One of the most basic but fundamental questions in the representation theory of Lie algebras is to prove complete reducibility for certain categories of modules. In the case of infinite-dimensional Lie algebras, this question is related to the representation theory of affine vertex algebras. In this talk we will discuss affine vertex algebras beyond admissible levels, where classical Lie theoretic methods are currently not sufficient for a complete determination of the category of weight modules. A level k is called collapsing if the simple affine W-algebra at level k associated with simple Lie algebra is isomorphic to its affine vertex subalgebra. We present a proof of the semi-simplicity of the Kazhdan-Lusztig category KL of affine vertex algebras at collapsing levels. The proof uses the representation theory of affine vertex algebras at collapsing levels and concepts from the theory of conformal embeddings.

The talk is motived by the series of joint papers with V. Kac, P. Moseneder-Frajria, P. Papi, O. Perse and I. Vukorepa.

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