Irreducible Representations of Finite Lie Conformal Algebras of Planar Galilean Type

It is well known that Galilean conformal algebras play important roles in the nonrelativistic anti-de Sitter/conformal field theory correspondence. The finite Lie conformal algebras $\mathfrak{P}\mathfrak{G}\left(a,b\right)$ of planar Galilean type can be viewed as Lie conformal analogues of certain planar Galilean conformal algebras. In this paper, we classify finite irreducible conformal modules over $\mathfrak{P}\mathfrak{G}\left(a,b\right)$ for all complex numbers a and b.

Introduction

Galilean conformal algebras have attracted much attention during the last decade (see, e.g. [[4], [5], [6], 23]) due to their importance in the nonrelativistic anti-de Sitter/conformal field theory correspondence conjecture. Recently, some Galilean conformal algebras with striking features have been studied from the pure Lie algebraic viewpoint (see, e.g. [2, 3, 19, 20]). A Lie conformal algebra is a fascinating generalization of a Lie algebra in the sense of pseudo-tensor category. So it is natural to study Galilean conformal algebras on the Lie conformal algebra level.

The notion of Lie conformal algebras, introduced by Kac [22], encodes the singular part of the operator product expansion of chiral fields in conformal field theory. The theory of finite simple/semisimple Lie conformal algebras has been greatly developed (see, e.g. [7, 12, 15, 16, 22, 27]). The smallest (but particularly important in physics) simple Lie conformal algebra is the Virasoro conformal algebra

$$\text{CVir}=ℂ[\partial ]L\text{with}\lambda \text{-bracket}\left[L_{\lambda }L\right]=\left(\partial +2\lambda \right)L.$$

It is natural to focus on non-semisimple Lie conformal algebras, especially those containing CVir as a subalgebra (see, e.g. [18, 24, 25] and the references therein).

The finite Lie conformai algebra $\mathfrak{P}\mathfrak{G}\left(a,b\right)$ of planar Galilean type introduced in [21] happens to be such an interesting Lie conformal algebra: it is not only a non-semisimple Lie conformal algebra containing Virasoro conformal subalgebra, but also has a Galilean conformal algebra background. Recall that $\mathfrak{P}\mathfrak{G}\left(a,b\right)$ with $a,b\in ℂ$ is defined as the free $ℂ[\partial ]$-module of rank four generated by {L, J, P, Q}, with following λ-brackets:

$$\left[L_{\lambda }L\right]=\left(\partial +2\lambda \right)L,$$

$$\left[L_{\lambda }J\right]=\left(\partial +\lambda \right)J,$$

$$\left[L_{\lambda }P\right]=\left(\partial +a\lambda +b\right)P,$$

$$\left[L_{\lambda }Q\right]=\left(\partial +a\lambda +b\right)Q,$$

$$\left[J_{\lambda }P\right]=Q,$$

$$\left[J_{\lambda }Q\right]=-P.$$

Other λ-brackets are given by the skew-symmetry or vanish. From the λ-bracket (1), we note that $\mathfrak{P}\mathfrak{G}\left(a,b\right)$ contains a Virasoro conformal subalgebra $ℂ[\partial ]L$. Other λ-brackets imply that $ℂ[\partial ]J\oplus ℂ[\partial ]P\oplus ℂ[\partial ]Q$ is a solvable ideal of $\mathfrak{P}\mathfrak{G}\left(a,b\right)$, and so $\mathfrak{P}\mathfrak{G}\left(a,b\right)$ is non-semisimple. Note also that the annihilation algebra associated to $\mathfrak{P}\mathfrak{G}\left(a,b\right)$ with (a, b) = (2, 0) is exactly corresponding to the planar Galilean conformal algebra introduced in [23] in the context of nonrelativistic conformal field theories.

One of the most important problems in the representation theory of Lie conformal algebras is to classify all their finite irreducible conformal modules. This problem has been solved for finite simple Lie conformal superalgebras [[9], [10], [11], [12], [13]] by a description of extremal vectors and degenerate modules. Recently, this problem has also been solved for infinite Lie conformal algebras of Block type [24] by a different technical method. In [21], we have classified the rank-one conformal modules over $\mathfrak{P}\mathfrak{G}\left(a,b\right)$. In this paper, by employing the techniques developed in [24], we give a complete classification of all finite irreducible conformal modules over $\mathfrak{P}\mathfrak{G}\left(a,b\right)$. Since conformal modules are in general not completely reducible, it is worth further considering the extension problem for these irreducible conformal $\mathfrak{P}\mathfrak{G}\left(a,b\right)$-modules, which will be solved in the near future work.

At the end of this paper, we discuss more potential directions for future investigation, and some potential applications of our results to physics from different view points, including comparison with the Euclidean group E(2) and density modules on the formal distribution Lie algebra of $\mathfrak{P}\mathfrak{G}\left(a,b\right)$.

This paper is arranged as follows. In Section 2, we first recall some definitions on Lie conformal algebras, and then present the Lie structures of the (extended) annihilation algebra of $\mathfrak{P}\mathfrak{G}\left(a,b\right)$. Then, in Section 3, we give a technical lemma on the representations of certain subquotient algebra of the annihilation algebra of $\mathfrak{P}\mathfrak{G}\left(a,b\right)$. In Section 4, we completely classify finite irreducible conformal modules over $\mathfrak{P}\mathfrak{G}\left(a,b\right)$ by showing that they must be free of rank one. Finally, in Section 5, we discuss some potential directions for future investigation, and some potential applications of our results to physics.