Exponential Equations Related to the Quantum 'ax + b' Group

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We study pairs b, β of unbounded selfadjoint operators, satisfying commutation rules inspired by the quantum 'ax + b' group [19]: bβ = -βb and β2= id except for kerb, on which β 2 = 0. We find all measurable, unitary-operator valued functions F satisfying the exponential equation: F(b, β)F(d, δ) = F((b, β) circled inside astrik sign (d, δ)), where d, δ satisfy the same commutation rules as b, β, and cirlcle astrik sign is modeled after the comultiplication of the quantum 'ax + b' group. This result is crucial for classification of all unitary representations of the quantum 'ax + b' group, which is achieved in our forthcoming paper.

Original languageEnglish (US)
Pages (from-to)419-453
Number of pages35
JournalCommunications in Mathematical Physics
Issue number3
StatePublished - Feb 2004
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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