Conformal Lie algebras via deformation theory

Abstract

We discuss possible notions of conformal Lie algebras, paying particular attention to graded conformal Lie algebras with $d$-dimensional space isotropy: namely, those with a $\mathfrak{co}(d)$ subalgebra acting in a prescribed way on the additional generators. We classify those Lie algebras up to isomorphism for all $d\geq 2$ following the same methodology used recently to classify kinematical Lie algebras, as deformations of the `most abelian' graded conformal algebra. We find 17 isomorphism classes of Lie algebras for $d\neq 3$ and 23 classes for $d=3$. Lie algebra contractions define a partial order in the set of isomorphism classes and this is illustrated via the corresponding Hesse diagram. The only metric graded conformal Lie algebras are the simple Lie algebras, isomorphic to either $\mathfrak{so}(d{+}1,2)$ or $\mathfrak{so}(d{+}2,1)$. We also work out the central extensions of the graded conformal algebras and study their invariant inner products. We find that central extensions of a Lie algebra in $d=3$ and two Lie algebras in $d=2$ are metric. We then discuss several other notions of conformal Lie algebras (generalised conformal, Schrödinger and Lifshitz Lie algebras) and we present some partial results on their classification. We end with some open problems suggested by our results.