Locally Convex Bialgebroid of an Action Lie Groupoid

Abstract

Action Lie groupoids are used to model spaces of orbits of actions of Lie groups on manifolds. For each such action groupoid M⋊H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M\\rtimes H$$\\end{document}, we construct a locally convex bialgebroid Dirac(M⋊H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathord {\ extrm{Dirac}}(M\\rtimes H)$$\\end{document} with an antipode over Cc∞(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathord {{\\mathcal {C}}^{\\infty }_{c}}(M)$$\\end{document}, from which the groupoid M⋊H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M\\rtimes H$$\\end{document} can be reconstructed as its spectral action Lie groupoid AGsp(Dirac(M⋊H))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathord {\\mathcal{A}\\mathcal{G}_{ sp }}(\\mathord {\ extrm{Dirac}}(M\\rtimes H))$$\\end{document}.