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The field of quantum \(GL(N,\pmb {\mathbb {C}})\) in the C\(^*\)-algebraic setting

  • Kenny De CommerEmail author
  • Matthias Floré
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Abstract

Given a unital \(*\)-algebra \(\mathscr {A}\) together with a suitable positive filtration of its set of irreducible bounded representations, one can construct a C\(^*\)-algebra \(A_0\) with a dense two-sided ideal \(A_c\) such that \(\mathscr {A}\) maps into the multiplier algebra of \(A_c\). When the filtration is induced from a central element in \(\mathscr {A}\), we say that \(\mathscr {A}\) is an s\(^*\)-algebra. We also introduce the notion of \(\mathscr {R}\)-algebra relative to a commutative s\(^*\)-algebra \(\mathscr {R}\), and of Hopf \(\mathscr {R}\)-algebra. We formulate conditions such that the completion of a Hopf \(\mathscr {R}\)-algebra gives rise to a continuous field of Hopf C\(^*\)-algebras over the spectrum of \(R_0\). We apply the general theory to the case of quantum \(GL(N,\mathbb {C})\) as constructed from the FRT-formalism.

Keywords

FRT quantum groups Quantized enveloping algebras Reflection equation algebra Locally compact quantum groups Deformation theory Continuous fields of C\(^*\)-algebras 

Mathematics Subject Classification

17B37 20G42 46L65 
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Notes

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Authors and Affiliations

  1. 1.Vakgroep WiskundeVrije Universiteit Brussel (VUB)BrusselsBelgium

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