Alexander Stolin (Chalmers University of Technology and Göteborg University)

Friday 24th March, 12.05-12.55pm, Carslaw 159

Irreducible highest-weight modules and equivariant quantization

The notion of deformation quantization, motivated by ideas coming from both physics and mathematics, was introduced in classical papers by Bayen, Flato, Fronsdal, Lichnerowicz, Sternheimer. The fact that any Poisson manifold can be quantized in this sense was proved by Kontzevich. However, finding exact formulas for specific cases of Poisson brackets is an interesting separate problem. There are several well-known examples of such explicit formulas. One of the first was the Moyal product quantizing the standard symplectic structure on a real even-dimensional space. Another one is the standard quantization of the Kirillov-Kostant-Souriau bracket on the dual space to a Lie algebra.

One of our main results is the connection between quantum dynamical twists and equivariant quantization. We give explicit formulas for star-products on certain subspaces of the algebra of regular functions on G/H, where G and H denote the Lie groups respectively associated with a simple Lie algebra and its Cartan subalgebra.