We describe some algebraic constructions (still somewhat mysterious) arising from solutions of the tt*-Toda equations, a system of p.d.e. related to the Toda equations and the Higgs equations. The solutions can be described using methods from the theory of integrable systems. This is based on the following articles: [GH] Martin Guest and Nan-Kuo Ho, Kostant, Steinberg, and the Stokes matrices of the tt*-Toda equations (arXiv: 1802.01126); [G] Martin Guest, Topological-antitopological fusion and the quantum cohomology of Grassmannians (arXiv:2012.01123); [GO] Martin Guest and Takashi Otofuji, Positive energy representations of affine algebras and Stokes matrices of the affine Toda equations (arXiv:2109.00765).