Howe-Moore type theorems for quantum groups and rigid \(C^{\ast}\)-tensor categories. (English) Zbl 1393.18003
The original Howe-Moore theorem referenced in the title of this paper (from [R. Howe and C. Moore, J. Funct. Anal. 32, 72–96 (1979; Zbl 0404.22015)]) states: for any connected non-compact simple Lie group with finite center, the matrix coefficients of every unitary representation without invariant vectors vanish at infinity. More generally, a locally compact group is said to have the Howe-Moore property if this property of matrix coefficients holds. Categories of unitary modules for locally compact groups are examples of rigid \(C^*\)-tensor categories: unitary modules have tensor products and duals, and homomorphisms have adjoints and norms, satisfying natural properties. Other examples of rigid \(C^*\)-tensor categories come from representation categories of quantum groups and subfactors.
In this paper, the authors give a definition of Howe-Moore property for general rigid \(C^*\)-tensor categories in terms of the vanishing at infinity of completely positive multipliers, which are certain complex-valued functions on the set of equivalence classes of irreducible objects introduced by S. Popa and S. Vaes, Comm. Math. Phys. 340, No. 3, 1239–1280 (2015; Zbl 1335.46055)]. Their main result is that a rigid \(C^*\)-tensor category has the Howe-Moore property if its fusion algebra is isomorphic to that of a connected compact simple Lie group with trivial center. Examples of rigid \(C^*\)-tensor categories satisfying this condition include unitary representations of \(q\)-deformations of connected compact simple Lie groups with trivial center, with \(q\in(0,1]\), as well as the category of \(M\)-bimodules associated with the Jones tower of a type-\(II_1\) subfactor \(N\subseteq M\) with Temperley-Lieb-Jones standard invariant \(\mathrm{TLJ}(\lambda)\) for \(\lambda^{-1}\geq 4\).
Additionally, the authors prove a characterization of central states on the quantum coordinate algebra of \(\mathrm{SU}_q(N)\) in terms of \(\mathrm{SU}(N)\)-bi-invariant positive-definite functions on \(\mathrm{SL}(N,\mathbb{C})\). Such central states coincide with the completely positive multipliers used to define the Howe-Moore property.
In this paper, the authors give a definition of Howe-Moore property for general rigid \(C^*\)-tensor categories in terms of the vanishing at infinity of completely positive multipliers, which are certain complex-valued functions on the set of equivalence classes of irreducible objects introduced by S. Popa and S. Vaes, Comm. Math. Phys. 340, No. 3, 1239–1280 (2015; Zbl 1335.46055)]. Their main result is that a rigid \(C^*\)-tensor category has the Howe-Moore property if its fusion algebra is isomorphic to that of a connected compact simple Lie group with trivial center. Examples of rigid \(C^*\)-tensor categories satisfying this condition include unitary representations of \(q\)-deformations of connected compact simple Lie groups with trivial center, with \(q\in(0,1]\), as well as the category of \(M\)-bimodules associated with the Jones tower of a type-\(II_1\) subfactor \(N\subseteq M\) with Temperley-Lieb-Jones standard invariant \(\mathrm{TLJ}(\lambda)\) for \(\lambda^{-1}\geq 4\).
Additionally, the authors prove a characterization of central states on the quantum coordinate algebra of \(\mathrm{SU}_q(N)\) in terms of \(\mathrm{SU}(N)\)-bi-invariant positive-definite functions on \(\mathrm{SL}(N,\mathbb{C})\). Such central states coincide with the completely positive multipliers used to define the Howe-Moore property.
Reviewer: Robert McRae (Nashville)
MSC:
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 20G42 | Quantum groups (quantized function algebras) and their representations |
| 46L10 | General theory of von Neumann algebras |
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