[Paper Review] Induced Representations of Quantum Groups

Oscar Arratia, M. A. del Olmo|ArXiv.org|Oct 24, 2001

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper presents a systematic construction of induced representations for quantum groups using a generalized Mackey method, focusing on bicrossproduct Hopf algebras with abelian kernels. By starting from one-dimensional characters of the commutative sector, it explicitly derives representations for two non-equivalent quantum deformations of the (1+1) Galilei algebra, demonstrating that the algebraic structure alone suffices for induction, while the coalgebra plays a supporting role in pairing computations.

ABSTRACT

In this paper we show how to construct explicitly induced representations for bicrossproduct Hopf algebras with abelian kernels starting from one-dimensional characters of the commutative sector. We introduce this technique by means of two concrete physical examples: two quantum deformations of the (1+1) Galilei algebra.

Motivation & Objective

• To develop a systematic method for constructing induced representations of quantum groups, particularly those with bicrossproduct structures.
• To extend Mackey’s theory of induced representations from Lie groups to quantum groups, focusing on the algebraic rather than corepresentation aspects.
• To apply the method to concrete physical examples: two non-equivalent quantum deformations of the (1+1) Galilei algebra.
• To investigate the role of the algebra versus coalgebra structures in the induction process.
• To lay the groundwork for a complete theory of induced representations in quantum group theory, including irreducibility, unitarity, and equivalence criteria.

Proposed method

• The method generalizes Mackey’s approach for semidirect product groups to bicrossproduct Hopf algebras, using one-dimensional characters of the commutative (abelian) kernel as starting data.
• Induced representations are constructed on the space of formal power series in a variable v, isomorphic to ℂ[[v]], representing the dual of the translation sector.
• The action of quantum group generators (H, P, K) on the induced module is computed via explicit formulas derived from the coproduct and adjoint action structure.
• Pairings between algebra and dual algebra elements are used to verify consistency, with duality relations given by ⟨K^m H^n P^p, v^q t^r x^s⟩ = m!n!p! δ^m_q δ^n_r δ^p_s.
• The coalgebra structure is used to compute tensor products and verify compatibility but is not essential for the induction mechanism itself.
• The construction relies on interpreting H and K as ladder operators on polynomials in v, enabling irreducibility analysis.

Experimental results

Research questions

• RQ1 How can Mackey’s method for induced representations be generalized to quantum groups with bicrossproduct structure?
• RQ2 What is the role of the algebraic structure versus the coalgebraic structure in the construction of induced representations?
• RQ3 Can induced representations of quantum groups be systematically constructed from one-dimensional characters of the commutative sector?
• RQ4 Under what conditions are the induced representations irreducible or unitary?
• RQ5 Can this method generate all irreducible representations of quantum groups, or are there limitations?

Key findings

• The induced representation for the standard quantum (1+1) Galilei algebra is constructed on the space ℂ[[v]], with explicit actions: φ(v) ⊲ K = φ′(v), φ(v) ⊲ P = φ(v)ia, and φ(v) ⊲ H = φ(v)[ib + (1/4ρ)(1 - e^{-4iaρ})v].
• For the non-standard quantum (1+1) Galilei algebra, the induced representation acts via: φ(v) ⊲ K = φ′(v), φ(v) ⊲ P = φ(v)ia, and φ(v) ⊲ H = φ(v)[ib + (1/4ρ)(1 - e^{-4iaρ})v], with the same functional form as the standard case.
• The representation labeled by (a,b) is equivalent to that labeled by (a,0), indicating that the parameter b can be absorbed via unitary equivalence.
• Irreducibility is established by showing that H and K act as ladder operators on the space of polynomials in v, generating the full module.
• The coalgebra structure is not essential for the induction process; only the algebraic relations and duality pairings are required.
• The method provides a systematic, non-ad hoc way to construct representations, contrasting with previous approaches that relied on case-specific constructions.