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Toward a Holographic Transform for the Quantum Clebsch-Gordan Formula
Shelburne, Ethan
Shelburne, Ethan
Abstract
The tensor product of two finite irreducible representations of sl(2,C) decomposes in the classical Clebsch-Gordan formula. In "Poisson and Fourier Transforms for Tensor Products," explicit intertwiners for the classical Clebsch-Gordan formula, known as Poisson and Fourier transforms, were introduced. When q is not a root of unity, the representation theory of the quantum group Uq(sl(2,C)) is analogous to that of sl(2,C) and finite irreducible representations decompose in the quantum Clebsch-Gordan formula. We introduce an algebraic approach to constructing an explicit holographic transform which is an intertwiner for the classical Clebsch-Gordan formula. We show this transform coincides with the Poisson transform up to a constant factor, which we compute. We then employ the same algebraic method to determine a holographic transform in the quantum case. Finally, we provide a conjectural form of a q-analogue to the Poisson transform and prove the form is correct in at least one special case.
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2021-05-01
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Mathematics