Positive Representations of Split Real Simply-laced Quantum Groups

Abstract

We construct the positive principal series representations for ${\mathcal{U}}_q({\mathfrak{g}}_{\mathbb{R}})$ where $\mathfrak{g}$ is of simply-laced type, parametrized by ${\mathbb{R}}_{\ge 0}^r$ where $r$ is the rank of $\mathfrak{g}$. We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group ${\mathbf{U}}_{\mathfrak{q}\tilde{\mathfrak{q}}}({\mathfrak{g}}_{\mathbb{R}})$ of the modular double and show that the representations of both parts of the modular double commute weakly with each other, there is an embedding into a quantum torus algebra, and the commutant contains its Langlands dual.

Finalized published version. Introduction has been rewritten to reflect recent progress and references added. Some typos fixed

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Cited by in corpus (13)

• Supergroup Structure of Jackiw-Teitelboim Supergravity
• Continuous tensor categories from quantum groups I: algebraic aspects
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• On tensor products of positive representations of split real quantum Borel subalgebra $U_{q\tilde{q}}(b_R)$
• On the continuous series for affine sl(2,R)
• Gauss-Lusztig Decomposition for $G{L}_q^{+}(N,R)$ and Representation by q-Tori
• Supersymmetry and the Modular Double
• On tensor product decomposition of positive representations of ${\mathcal{U}}_{q\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))$
• Positive representations, multiplier Hopf algebra, and continuous canonical basis
• Positive Casimir and Central Characters of Split Real Quantum Groups
• Parabolic Positive Representations of ${\mathcal{U}}_q({\mathfrak{g}}_{\mathbb{R}})$
• On explicit realization of algebra of complex powers of generators of $U_q(\mathfrak{sl}(3))$