Abstract
In this work, we focus on describing the space of bi-invariant metrics in a Lie group up to isometry. i.e, metrics invariant under both left and right translations. We show that BI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {BI}$$\\end{document}, the moduli space of bi-invariant metrics, is an orbifold. Moreover, we give an explicit description of this orbifold, and of EBI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {EBI}$$\\end{document}, the space of bi-invariant metrics equivalent under isometries and scalar multiplies.
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