The Null Decomposition of Conformal Algebras

TL;DR

This paper investigates how the enveloping algebra of conformal algebras decomposes relative to the mass-squared operator, revealing a structure involving Poincare subalgebras and vector operators, with detailed cases for 2D and 3D.

Contribution

It provides a detailed analysis of the null decomposition of conformal algebras in arbitrary dimensions, including explicit constructions of Casimir operators for low dimensions.

Findings

The subalgebra commuting with the mass-squared operator is generated by Poincare subalgebra and a vector operator.

Explicit Casimir operators are constructed for 2D and 3D conformal algebras.

The structure of the algebra simplifies in special low-dimensional cases.

Abstract

We analyze the decomposition of the enveloping algebra of the conformal algebra in arbitrary dimension with respect to the mass-squared operator. It emerges that the subalgebra that commutes with the mass-squared is generated by its Poincare subalgebra together with a vector operator. The special cases of the conformal algebras of two and three dimensions are described in detail, including the construction of their Casimir operators.