# On the boundary Carrollian conformal algebra

## TL;DR

This paper explores the mathematical structure of the boundary Carrollian conformal algebra (BCCA), constructing modules, analyzing irreducibility, and developing new representations like Whittaker modules to deepen understanding of this infinite-dimensional Lie algebra.

## Contribution

It introduces the first detailed study of BCCA modules, irreducibility criteria, and constructs new Whittaker modules, advancing the mathematical understanding of this algebra.

## Key findings

- Constructed modules for BCCA and subalgebras from known modules.
- Established irreducibility criteria for BMS$_3$ and BCCA modules.
- Developed new Whittaker modules and proved their irreducibility.

## Abstract

We initiate the mathematical study of the boundary Carrollian conformal algebra (BCCA), an infinite-dimensional Lie algebra recently discovered in the context of Carrollian physics. The BCCA is an intriguing object from both physical and mathematical perspectives, since it is a filtered but not graded Lie algebra. In this paper, we first construct some modules for the BCCA and one of its subalgebras, which we call $\mathcal{O}$, by restriction of well-known modules of the BMS$_3$ and Witt algebras respectively. Along the way, we prove the irreducibility criteria for the so-called ``induced modules'' of the BMS$_3$ algebra (which we prefer to call massive modules to avoid ambiguity) and show that this is the same criteria for the irreducibility of the Verma modules of the BMS$_3$ algebra. Interestingly, the modules generated by the action of the BCCA on the generating vector of the massive modules are also irreducible under the same criteria. When this criteria holds, every massive module decomposes into a direct sum of two BCCA-submodules, each of which we conjecture to be irreducible. Meanwhile, restricting Verma modules to the BCCA and $\mathcal{O}$ leads to free or ``almost free'' modules, which are not particularly interesting from a representation-theoretic viewpoint. This motivates the construction of BCCA modules intrinsically. To do this, we go through some structure theory on the BCCA to define a new basis and a decreasing filtration on the algebra, using which we construct Whittaker modules over the BCCA and the subalgebra $\mathcal{O}$ and prove criteria for their irreducibility.