Formal loops, Tate objects and tangent Lie algebras

If $M$ is a symplectic manifold then the space of smooth loops $\mathrm C^{\infty}(\mathrm S^1,M)$ inherits of a quasi-symplectic form. We will focus in this thesis on an algebraic analogue of that result. Kapranov and Vasserot introduced and studied the formal loop space of a scheme $X$. It is an algebraic version of the space of smooth loops in a differentiable manifold. We generalize their construction to higher dimensional loops. To any scheme $X$ -- not necessarily smooth -- we associate $\mathcal L^d(X)$, the space of loops of dimension $d$. We prove it has a structure of (derived) Tate scheme -- ie its tangent is a Tate module: it is infinite dimensional but behaves nicely enough regarding duality. We also define the bubble space $\mathcal B^d(X)$, a variation of the loop space. We prove that $\mathcal B^d(X)$ is endowed with a natural symplectic form as soon as $X$ has one. To prove our results, we develop a theory of Tate objects in a stable $(\infty,1)$-category $\mathcal C$. We also prove that the non-connective K-theory of $\mathbf{Tate}(\mathcal C)$ is the suspension of that of $\mathcal C$. The last chapter is aimed at a different problem: we study there the existence of a Lie structure on the tangent of a derived Artin stack. This in particular applies to not necessarily smooth schemes. Throughout this thesis, we will use the tools of $(\infty,1)$-categories and symplectic derived algebraic geometry.

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