[Paper Review] Some representations of planar Galilean conformal algebra

N. Aizawa|arXiv (Cornell University)|Dec 27, 2012

Algebraic structures and combinatorial models5 references3 citations

TL;DR

This paper investigates the representation theory of the planar Galilean conformal algebra (GCA) in (2+1) dimensions with central extensions. It proves that Verma modules over the centrally extended algebra are irreducible when the highest weights are non-vanishing, using explicit computation of the Kac determinant. The study further develops coadjoint representations of the algebra and its associated Lie group, providing a concrete example of a coadjoint orbit and laying groundwork for future classification of orbits and physical applications.

ABSTRACT

Representation theory of an infinite dimensional Galilean conformal algebra introduced by Martelli and Tachikawa is developed. We focus on the algebra defined in (2+1) dimensional spacetime and consider central extension. It is then shown that the Verma modules are irreducible for non-vanishing highest weights. This is done by explicit computation of Kac determinant. We also present coadjoint representations of the Galilean conformal algebra and its Lie group. As an application of them, a coadjoint orbit of the Galilean conformal group is given in a simple case.

Motivation & Objective

• To investigate the representation theory of the infinite-dimensional Galilean conformal algebra in (2+1) spacetime dimensions.
• To determine whether the algebra admits an exotic central extension that makes its Abelian ideal non-Abelian.
• To analyze highest weight representations, particularly Verma modules, and establish their irreducibility.
• To construct coadjoint representations of the Lie algebra and its associated Lie group.
• To provide a concrete example of a coadjoint orbit of the Galilean conformal group.

Proposed method

• The paper introduces central extensions to the planar Galilean conformal algebra, adding central charges α and β to the Virasoro and U(1) current subalgebras.
• It employs a triangular decomposition of the algebra to define highest weight vectors and construct Verma modules.
• The Kac determinant is explicitly computed to analyze the irreducibility of Verma modules.
• The regular dual space of the Lie algebra is used to define coadjoint representations.
• The coadjoint action of the Lie algebra and its group is derived using the standard group action formula on the dual space.
• A specific example of a coadjoint orbit is constructed by combining the coadjoint actions of the Virasoro subgroup and the (ξ, η₁, η₂)-subgroup.

Experimental results

Research questions

• RQ1 Does the planar Galilean conformal algebra in (2+1) dimensions admit an exotic central extension that non-abelianizes the Pₙⁱ ideal?
• RQ2 Under what conditions are Verma modules over the centrally extended Galilean conformal algebra irreducible?
• RQ3 How can coadjoint representations of the infinite-dimensional Galilean conformal algebra and its group be explicitly constructed?
• RQ4 What is the structure of a coadjoint orbit for the Galilean conformal group in a simple case?
• RQ5 Can the Galilean conformal group act on a space of differential operators, and if so, what is the nature of such an action?

Key findings

• The Verma modules of the centrally extended planar Galilean conformal algebra are irreducible when the highest weights are non-vanishing, as confirmed by the non-vanishing Kac determinant.
• The algebra does not admit an exotic central extension; instead, it supports standard central extensions for the Virasoro and U(1) current subalgebras with central charges α and β.
• The coadjoint action of the Virasoro subgroup on the dual space is given by the standard transformation law involving the Schwarzian derivative.
• The coadjoint action of the (ξ, η₁, η₂)-subgroup is explicitly computed, involving rotations and shifts in the dual components.
• A specific coadjoint orbit is constructed by combining the actions of the Virasoro and (ξ, η₁, η₂) subgroups, illustrating the structure of the orbit space.
• The results lay a foundation for the full classification of coadjoint orbits and suggest potential physical applications via group actions on differential operators.