[Paper Review] On some low dimensional quantum groups

W. Pusz, Piotr M. Sołtan|ArXiv.org|Dec 20, 2007

Advanced Operator Algebra Research20 references3 citations

TL;DR

This paper constructs and analyzes low-dimensional quantum groups as deformations of the affine 'az+b' and 'ax+b' groups using C*-algebraic and operator algebraic methods. It classifies three types of quantum deformations based on the parameter q, establishes comultiplications via multiplicative unitaries, and demonstrates that quantum 'ax+b' groups can be reduced to finite-size versions at roots of unity, with the minimal size being 2.

ABSTRACT

This paper is an adaptation of a chapter from an upcoming monograph on noncommutative geometry and quantum groups. We present examples of non compact quantum groups which are deformations of low dimensional Lie groups. The paper is of expository nature and provides both particular examples and some general procedures for constructing them.

Motivation & Objective

• To construct and classify low-dimensional quantum groups as deformations of the affine 'az+b' and 'ax+b' groups.
• To address technical challenges in realizing commutation relations via unbounded operators on Hilbert spaces.
• To establish C*-bialgebra structures with comultiplications via multiplicative unitaries.
• To analyze the size of quantum 'ax+b' groups and show that finite-size versions arise when the deformation parameter is a root of unity.
• To demonstrate that the minimal size of a quantum 'ax+b' group is 2, achieved when the parameter s is a root of unity.

Proposed method

• Uses a C*-algebraic approach to realize quantum deformations of the 'az+b' and 'ax+b' groups by defining generators a, b, β, w with specific commutation relations.
• Employs a Hopf *-algebra structure with comultiplication Δ(a) = a⊗a and Δ(b) = a⊗b + b⊗I on the algebraic level.
• Constructs a multiplicative unitary operator W on L²(ℝ×S¹)⊗L²(ℝ×S¹) satisfying the pentagon equation W₁₂W₁₃ = W₂₃W₁₂W₂₃*.
• Applies the quantum double construction to obtain a quantum deformation of GL(2,ℂ) from a quantum 'az+b' group.
• Uses the Zakrzewski relation to show that log(π(a)) and log(π(|b|)) satisfy canonical commutation relations, implying irreducibility via the Stone–von Neumann theorem.
• Reduces the size of quantum 'ax+b' groups by quotienting the C*-algebra A by the ideal generated by w^N − I when s^N = 1, yielding a finite-size quantum group of size 2N.

Experimental results

Research questions

• RQ1 How can quantum deformations of the 'az+b' and 'ax+b' groups be constructed at the C*-algebra level despite challenges in realizing unbounded operators?
• RQ2 What are the three distinct types of quantum deformations of the 'az+b' group, and how do they depend on the value of the deformation parameter q?
• RQ3 How does the multiplicative unitary operator W encode the comultiplication and satisfy the pentagon equation in the quantum 'ax+b' group construction?
• RQ4 What determines the size of a quantum 'ax+b' group, and how does it change when the deformation parameter becomes a root of unity?
• RQ5 Can the quantum double construction be used to obtain quantum deformations of larger groups such as GL(2,ℂ) and the Lorentz group from the 'az+b' quantum groups?

Key findings

• The paper identifies three types of quantum 'az+b' deformations—(I), (II), and (III)—based on the subgroup Γ_q generated by q, with distinct spectral and geometric structures.
• For the quantum 'az+b' group, the C*-bialgebra (A, Δ) is constructed using a multiplicative unitary W satisfying the pentagon equation, ensuring associativity of the comultiplication.
• The quantum 'ax+b' group is shown to be of infinite size in general, as its representations are multiples of the unique irreducible representation of the canonical commutation relations.
• When the deformation parameter q² = e^(-iħ) is a root of unity, the quantum 'ax+b' group can be quotiented to yield a finite-size quantum group of size 2N, where N is the smallest integer such that s^N = 1.
• The minimal possible size of a quantum 'ax+b' group is 2, achieved when s is a primitive root of unity of order 2, corresponding to the known 'old' quantum 'ax+b' groups.
• The quantum double construction applied to a quantum 'az+b' group yields a quantum deformation of GL(2,ℂ), and is also used to construct two distinct quantum deformations of the Lorentz group.