Global symmetries of Yang-Mills squared in various dimensions

Tensoring two on-shell super Yang-Mills multiplets in dimensions D ≤ 10 yields an on-shell supergravity multiplet, possibly with additional matter multiplets. Associating a (direct sum of) division algebra(s) D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{D} $$\end{document} with each dimension 3 ≤ D ≤ 10 we obtain a formula for the supergravity U-duality G and its maximal compact subgroup H in terms of the internal global symmetry algebras of each super Yang-Mills theory. We extend our analysis to include supergravities coupled to an arbitrary number of matter multiplets by allowing for non-supersymmetric multiplets in the tensor product.