# The bubble algebras at roots of unity

## TL;DR

This paper introduces multi-colour partition algebras and bubble algebras, providing techniques to analyze their structure at roots of unity, especially in non-semisimple cases, advancing algebraic understanding.

## Contribution

The paper defines new multi-colour partition and bubble algebras and develops methods to determine their structure over complex numbers at roots of unity.

## Key findings

- Defined multi-colour partition algebras and bubble algebras.
- Developed techniques for structural analysis in non-semisimple cases.
- Extended understanding of algebraic properties at roots of unity.

## Abstract

We introduce multi-colour partition algebras $P_{n,m}(\delta_0, ..., \delta_{m-1})$, which are generalization of both bubble algebras and partition algebras, then define the bubble algebra $T_{n,m}(\delta_0, ..., \delta_{m-1})$ as a sub-algebra of the algebra $P_{n,m}(\delta_0, ..., \delta_{m-1})$. We present general techniques to determine the structure of the bubble algebra over the complex field in the non-semisimple case.