Heisenberg and Weil representations in noncommutative geometry

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project aims to define and apply non-commutative analogues of the Heisenberg and Weil representations of the symplectic group acting on integrable complex-valued functions on finite-dimensional real vector spaces. The noncommutative analogues should replace the commutative algebra of complex-valued functions on n-dimensional real space by a free algebra on n generators, or more generally by path algebras of quivers. This should extend a construction of a Lie algebra action found earlier by the co-PI and Ginzburg. These representations should be useful in studying Calabi-Yau potentials for completed group algebras of fundamental groups of compact aspherical three-manifolds. More generally, this project aims to understand "non-commutative quantization" in the sense of replacing an associative algebra whose variety of n-dimensional representations is Poisson by some structure which maps under a generalization of the representation functor to a quantization of the original Poisson variety. This construction would include the non-commutative differential operators on the associative algebra, which are used in the definition of the Heisenberg representation. Finally, the project aims to continue studying potentials for Calabi-Yau algebras in three and higher dimensions, such as Sklyanin algebras, and apply this to their deformation theory and the deformation theory of singular Poisson varieties, generalizing works of the co-PI, as well as of Ginzburg, Etingof, and others. Various aspects of the problem will be approached using computational algebra and are suitable as projects for undergraduate students. In less technical language, the project aims to connect quantization theory (e.g., quantum mechanics and symmetries) with representation theory (the study of ways that algebras can act on vector spaces), by studying objects whose representations are themselves quantizations. This should have applications to understanding the structure of three-dimensional spaces as well as deformations of various well-known classical and quantum systems. An important part involves the study of algebras related to string theory which are defined by certain graphs and their relations, and particularly those spaces related to Calabi-Yau geometry. A central example is the Weil representation, which incorporates the Fourier transform between position and momentum coordinates, or time and frequency coordinates, as well as all in-between coordinates.