Title: The bubble algebras 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.