Derived geometry and higher quantum groups

This thesis presents a novel path to the explicit construction of models of `higher quantum groups' by studying positively-shifted Poisson structures on a generic semi-free commutative differential graded algebra and investigating exactly how they induce infinitesimal deformations of the symmetric monoidal structure on the dg-category of semi-free dg-modules over such a commutative differential graded algebra. First we develop a graphical calculus to characterise the shifted Poisson structures on finitely generated semi-free commutative differential graded algebras in terms of families of homotopy-coherent data adjoined to the corresponding finite-dimensional higher Lie algebra. Such a characterisation allows us to show that we recover the notion of higher Lie quasibialgebra due to Bai, Sheng and Zhu. When applied to an ordinary Lie algebra, we recover Safronov's result that the 1 and 2-shifted Poisson structure in this case are given by quasi-Lie bialgebra structures and invariant symmetric 2-tensors, respectively. We generalise these results to the case of a Lie 2-algebra and obtain 1, 2, 3 and 4-shifted Poisson structures which we interpret as semiclassical data of higher quantum groups. In particular, if we assume a 2-term differential graded Lie algebra and trivialise part of the data of a 2-shifted Poisson structure then we recover a special case of the notion of a symmetric quasi-invariant tensor in a differential crossed module due to Cirio and Faria Martins. It is then shown that every 2-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra $A$ defines a concrete infinitesimal 2-braiding on the homotopy 2-category of the symmetric monoidal dg-category of finitely generated semi-free dg-modules over A. This provides a concrete realisation, to first order in the deformation parameter, of the abstract deformation quantisation results in derived algebraic geometry due to Calaque, Pantev, Toën, Vaquié and Vezzosi. Of particular interest is the case when A is the Chevalley-Eilenberg algebra of a higher Lie algebra, where the braided monoidal deformations developed in this thesis may be interpreted as candidates for representation categories of higher quantum groups. Upon choosing the usual ansatz of the braiding and associator as, respectively, the exponential and Drinfeld's Knizhnik-Zamolodchikov series of the infinitesimal 2-braiding, it is demonstrated that the hexagon axioms are obstructed at second order by modifications. These modifications are shown to satisfy the requisite axioms of a braided monoidal 2-category (i.e., the modifications are shown to be `infinitesimal hexagonators') provided that the infinitesimal 2-braiding is totally y-equivariant and coherent in Cirio and Faria Martins' sense. We show that those infinitesimal 2-braidings induced by 2-shifted Poisson structures are indeed totally y-equivariant and we conjecture that coherency also holds by relating the condition to the third-weight component of the Maurer-Cartan equation that a 2-shifted Poisson structure definitively satisfies. We then demonstrate that the pentagonator is nontrivial by determining its third-order term. Finally, we study the problem of sylleptic deformations and show that a 3-shifted Poisson structure induces ``infinitesimal syllepses" which can be trivially integrated to all orders. We also show that a ``coboundary" 2-shifted Poisson structure induces ``infinitesimal coboundary syllepses" and these admit of a Cartier integration alongside that of infinitesimal 2-braidings and we carry this out to second and third order.