Interactions between representation theory, Poisson algebras and differential algebraic geometry

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Representation theory is one of the most active fields of mathematics today with applications to many of the sciences and interactions with many other mathematical disciplines such as number theory, combinatorics, geometry, probability theory, quantum mechanics and quantum field theory. This beautiful subject originated in a letter to Frobenius by Dedekind. Roughly speaking, the idea is to study algebras through their symmetries. Despite many successes and applications, many basic questions remain challenging. For instance, it is often quite difficult (if not impossible) to classify the irreducible representations of a given algebra. A now standard approach to this problem, proposed by Dixmier, is to study the annihilators of the irreducible representations, the so-called primitive ideals. Classifying primitive ideals of an algebra can be seen as a first approximation towards understanding the representation theory of the algebra. In the case of enveloping algebras of finite dimensional complex Lie algebras, Dixmier and Moeglin proved that, among the prime ideals, primitive ideals can be characterized both algebraically and topologically. These algebraic and topological criteria also characterise primitive ideals among prime ideals in many other algebras, in which case we say that the Dixmier-Moeglin equivalence holds. Poisson algebras first appeared in the work of Poisson two centuries ago when he was studying the three-body problem in celestial mechanics. Since then, Poisson algebras have been shown to be connected to many areas of mathematics and physics, and so, because of their wide range of applications, their study is of great interest for both mathematicians and theoritical physicists. Currently, this subject is one of the most active in both mathematics and mathematical physics. One way to approach Poisson algebras is via quantisation. In physics, quantisation is the transition from classical to quantum mechanics. Mathematically, (deformation) quantisation is the transition from Poisson algebras/geometry to noncommutative algebras/geometry. In the context of deformation quantisation, Poisson algebras are the semiclassical limits of noncommutative algebras. Roughly speaking, the noncommutative algebraic geometry of the ``quantum'' spaces is closely related to the geometry of the space of symplectic leaves. In the spirit of deformation quantisation, one is led to study a Poisson analogue of the Dixmier-Moeglin equivalence, the so-called Poisson Dixmer-Moeglin Equivalence. The Poisson Dixmer-Moeglin Equivalence was established for affine Poisson algebras with suitable torus actions by Goodearl, and for Poisson algebras with only finitely many Poisson primitive ideals by Brown and Gordon. Given these successes, Brown and Gordon asked in 2002 whether the Poisson Dixmer-Moeglin Equivalence holds for all affine complex Poisson algebras. In a recent paper with Bell, Leon Sanchez and Moosa, we completely answered this question thanks to a novel approach based on tools from differential algebraic geometry and the model theory of differential fields. This project arises from my desire to continue this new line of research at the crossroad between Poisson geometry and representation theory on one hand, and differential algebraic geometry and model theory on the other hand. This highly novel approach of Poisson geometry/representation theory has already led to solving a 12-year old question of Brown and Gordon. As always, linking different areas of mathematics will be the source of deep results. The aim of this project is to further study this new approach of Poisson geometry/representation theory via differential algebraic geometry. This will lead to progress in the representation theory of Hopf algebras, twisted homogeneous coordinate rings and Poisson algebras, as well as to new tools to study algebraic D-varieties.