[Paper Review] Examples of locally compact quantum groups through the bicrossed product construction

Stefaan Vaes|ArXiv.org|Sep 6, 2000

Advanced Operator Algebra Research12 references5 citations

TL;DR

This paper constructs explicit examples of non-compact locally compact quantum groups using the bicrossed product construction from matched pairs of locally compact groups. It demonstrates that the resulting quantum groups arise as crossed products of von Neumann algebras, with one example yielding the universal enveloping algebra of the Heisenberg Lie algebra and its dual reflecting the Heisenberg group's multiplication law, thereby illustrating duality in quantum groups.

ABSTRACT

The main aim of this paper is to introduce some examples of non-compact locally compact quantum groups to a non-specialized audience. The major importance of these examples is their simplicity. Other examples as the quantum E(2) group of Woronowicz are much more difficult to construct. We will make use of the bicrossed product construction to obtain our examples. This construction dates back to Kac, Takeuchi and Majid. It was also considered by Baaj & Skandalis.

Motivation & Objective

• To provide accessible, concrete examples of non-compact locally compact quantum groups for a non-specialized audience.
• To demonstrate the bicrossed product construction as a systematic method for generating such quantum groups.
• To illustrate how quantum groups emerge naturally from matched pairs of groups, particularly in the context of duality and harmonic analysis.
• To motivate the study of locally compact quantum groups by showing their relevance in quantum symmetries and operator algebraic duality.

Proposed method

• The bicrossed product construction is applied to matched pairs of locally compact groups (G, H), where actions α and β satisfy compatibility conditions.
• The comultiplication Δ on the von Neumann algebra M is defined via a unitary W on L²(G×H×G×H), given by (Wξ)(g,s,h,t) = ξ(β_{α_g(s)^{-1}t}(h)g, s, h, α_g(s)^{-1}t).
• The resulting quantum group (M, Δ) is shown to admit a left-invariant weight, confirming its structure as a locally compact quantum group.
• Concrete examples are constructed using G = (ℝ, +) and H = {(a,b) | a>0, b∈ℝ} with specific actions α and β defined piecewise.
• Formal computations yield a Hopf *-algebra structure on the generators A, B, C, with relations [A,C] = B, B central, and comultiplication Δ(A) = A⊗A, Δ(B) = A⊗B + B⊗A⁻¹, Δ(C) = C⊗A⁻² + 1⊗C.
• The dual quantum group is obtained by swapping roles of G and H, yielding a dual Hopf *-algebra with different relations and comultiplication.

Experimental results

Research questions

• RQ1 How can the bicrossed product construction be used to generate non-compact locally compact quantum groups from matched pairs of groups?
• RQ2 What is the explicit structure of the von Neumann algebra and comultiplication in such quantum groups?
• RQ3 How do the formal Hopf *-algebra structures on generators relate to classical Lie groups like the Heisenberg group?
• RQ4 In what way does the duality between the quantum group and its dual manifest in the algebraic relations and comultiplication?

Key findings

• The bicrossed product construction yields a locally compact quantum group (M, Δ) from a matched pair (G, H), with M a crossed product von Neumann algebra acting on L²(G×H).
• For G = (ℝ, +) and H = {(a,b) | a>0, b∈ℝ} with defined actions, the construction produces a quantum group whose formal Hopf *-algebra has generators A, B, C satisfying [A,C] = B, B central, and Δ(A) = A⊗A, Δ(B) = A⊗B + B⊗A⁻¹, Δ(C) = C⊗A⁻² + 1⊗C.
• The dual quantum group arises from swapping G and H, yielding a dual Hopf *-algebra with generators satisfying [A,B] = 2B, [A,C] = 2C, [B,C] = C², and comultiplication Δ(A) = A⊗1 + 1⊗A, Δ(B) = B⊗1 + 1⊗B + A⊗C, Δ(C) = C⊗1 + 1⊗C.
• The algebraic structure of the dual quantum group precisely reflects the multiplication law of the Heisenberg group: (a,b,c)·(a′,b′,c′) = (a+a′, b+b′+ac′, c+c′).
• The construction confirms that the quantum group and its dual are linked via duality, with the dual weight on the crossed product realizing left-invariance.