Functional equations of axiomatic multiple Dirichlet series, Weyl groupoids, and quantum algebra

We prove functional equations for multiple Dirichlet series defined by a collection of five geometric axioms. We find functional equations of two types: one modeled on the functional equations of Dirichlet $L$-functions, and another modeled on the functional equations of Kubota $L$-series with Gauss sums as coefficients. These functional equations generate groupoid structures, which we relate to the Weyl groupoids of arithmetic root systems. From the known classification of arithmetic root systems, we obtain a complete classification of multiple Dirichlet series which can be used to compute moments of $L$-functions via established analytic techniques. Our classification includes all moments of $L$-functions which have appeared in the multiple Dirichlet series literature previously, alongside some new moments. Finally, we give applications of our functional equations to quantum algebra, specifically the cohomology of Nichols algebras.