Some Properties of Planar Galilean Conformal Algebras

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.

Galilean conformal algebras in two spatial dimension

“…The latter is a consequence of our explicit formula of Kac determinant which is also our main result. Some of the preliminary results (for the simplest member of the algebras defined in (2 + 1) dimension) have already been reported elsewhere [44,45].…”

Representations of Generalized Loop Planar Galilean Conformal Algebras W(Γ)

“…The planar Galilean conformal algebra was first introduced by Bahturin and Gopakumar in reference [1]. In Section 2 of reference [4], the Galilean conformal algebra of finite dimension in the d spatial dimension was introduced. For a given d, Galilean conformal algebra is labelled by a half-integer l. The infinite dimension d = 2, l = 1 is planar Galilean conformal algebra.…”

Whittaker modules for the planar Galilean conformal algebra and its central extension

“…The planar Galilean conformal algebra G, which was first introduced by Bagchi and Gopakumar in [4] and named by Aizawa in [2], is a Lie algebra with a basis {L m , H m , I m , J m | m ∈ Z} and the nontrivial Lie brackets defined by…”