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 $(\mathrm{\infty },1)$-category $\mathcal{C}$. We also prove that the non-connective K-theory of $\mathbf{T}\mathbf{a}\mathbf{t}\mathbf{e}(\mathcal{C})$ 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 $(\mathrm{\infty },1)$-categories and symplectic derived algebraic geometry.