The Time-Frequency Covariance Principle on Unimodular Kac Algebras

Abstract

This paper extends the short-time Fourier transform (STFT), a fundamental tool in time-frequency analysis, to the quantum group setting of unimodular Kac algebras. For a unimodular Kac algebra \mathbb{G}, we introduce a time-frequency shift operator that combines left translation and modulation operators. Using a window vector in the Hilbert space L^2(\mathbb{G}), we define the corresponding STFT and establish its essential analytic properties, including a Plancherel theorem, the Moyal identity, an inversion formula, and a fundamental identity. Furthermore, we explore the projective corepresentation structure of the time-frequency shift operator, and prove that its reflected version induces a continuous projective left representation of the dual quantum group of the quantum double. Finally, we derive the covariance principle and several uncertainty principles.