We investigate the phase space symmetries and conserved charges of
    homogeneous gravitational minisuperspaces. These (0+1)-dimensional reductions
    of general relativity are defined by spacetime metrics in which the dynamical
    variables depend only on a time coordinate, and are formulated as mechanical
    systems with a non-trivial field space metric (or supermetric) and effective
    potential. We show how to extract conserved charges for those minisuperspaces
    from the homothetic Killing vectors of the field space metric. In the case of
    two-dimensional field spaces, we exhibit a universal 8-dimensional symmetry
    algebra given by the semi-direct sum of
    $\mathfrak{sl}(2,\mathbb{R})\oplus\mathbb{R}$ with the two-dimensional
    Heisenberg algebra $\mathfrak{h}_2\simeq\mathbb{R}^4$. We apply this to the
    systematic study of the Bianchi models for homogeneous cosmology. This extends
    previous results on the $\mathfrak{sl}(2,\mathbb{R})$ algebra for
    Friedmann-Lemaitre-Robertson-Walker cosmology, and the Poincar\’e symmetry for
    Kantowski-Sachs metrics describing the black hole interior. The presence of
    this rich symmetry structure already in minisuperspace models opens new doors
    towards quantization and the study of solution generating mechanisms.

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