University of Sydney Algebra Seminar

Marcello Lanfranchi

Friday 18 September, 12-1pm, in Carslaw 275

The Lie Group-Lie Algebra Correspondence in Tangent Categories

Classic Lie theory establishes a correspondence between Lie groups and Lie algebras. An analogous correspondence also exists for group objects in affine schemes. Both smooth manifolds and affine schemes form tangent categories, which provide a categorical context for differential geometry. In this talk, we show how to construct the Lie correspondence entirely from the tangent structure. We define Lie groups in a tangent category as group objects that admit the tangent space at the unit and construct the internal Lie algebra as the representing object of a certain functor. Lastly, we show that our construction generalizes the usual Lie correspondence in both differential and algebraic geometry. During the talk, I will give a quick introduction to tangent categories. Paper: https://arxiv.org/abs/2609.03449