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@PHDTHESIS{Braun:1013585,
      author       = {Braun, Tobias},
      othercontributors = {Nebe, Gabriele and Robertz, Daniel},
      title        = {{C}lifford orders},
      school       = {RWTH Aachen University},
      type         = {Dissertation},
      address      = {Aachen},
      publisher    = {RWTH Aachen University},
      reportid     = {RWTH-2025-05622},
      pages        = {1 Online-Ressource : Illustrationen},
      year         = {2025},
      note         = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
                      University; Dissertation, RWTH Aachen University, 2025},
      abstract     = {The aim of this thesis is to investigate the properties of
                      the Clifford algebra of a quadratic lattice over a Dedekind
                      domain and its completions and to compare it with the
                      properties of the Clifford algebra of its ambient quadratic
                      space. The new object that arises this way - the Clifford
                      order - has not yet been studied extensively as an
                      independent object. The present thesis addresses this, using
                      both the theory of orders and Clifford algebras to extend
                      well-known results that hold for Clifford algebras over
                      fields to this new, more general setting. It was long known
                      to theory that the centraliser of the even Clifford algebra,
                      the so-called centroid is a cornerstone for describing the
                      Clifford algebra of an orthogonal direct sum of quadratic
                      spaces. This thesis develops the theory of quadratic orders,
                      to describe the centroids of Clifford orders on an abstract
                      level. In this context, a new invariant of a quadratic
                      lattice, the quadratic discriminant, is introduced, allowing
                      for a simplified computation of the centroids in certain
                      situations. As applications, the centroids of the maximal
                      lattices over a Dedekind domain and of an arbitrary root
                      lattice are computed, and an effective way to determine the
                      Clifford order of the orthogonal direct sum of two quadratic
                      lattices is presented. Additionally, an algorithm to compute
                      the centroid of a given Clifford orders over an arbitrary
                      Dedekind domain is described. Finally, this thesis
                      classifies the Clifford orders and the centroids of all
                      maximal lattices over a complete discrete valuation ring and
                      describes them as a subalgebra of their ambient Clifford
                      algebra.},
      cin          = {114820 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)114820_20140620$ / $I:(DE-82)110000_20140620$},
      pnm          = {DFG project G:(GEPRIS)286237555 - TRR 195: Symbolische
                      Werkzeuge in der Mathematik und ihre Anwendung (286237555)},
      pid          = {G:(GEPRIS)286237555},
      typ          = {PUB:(DE-HGF)11},
      doi          = {10.18154/RWTH-2025-05622},
      url          = {https://publications.rwth-aachen.de/record/1013585},
}