Symmetries of blob algebras: Derivations, outer classes and cellular transfer

We study derivations of the blob algebra ${\mathsf{B}}_n(q,\delta ,\kappa )$ over a field of characteristic zero and their interaction with its Graham–Lehrer cellular structure. From the defining presentation, we obtain a necessary and sufficient extension criterion that translates the Leibniz rule into explicit compatibility conditions on the generators. When the algebra is semisimple, every derivation is inner, and hence $\mathrm{D}\mathrm{e}\mathrm{r}({\mathsf{B}}_n)=\mathrm{I}\mathrm{n}\mathrm{n}({\mathsf{B}}_n)\cong {\mathsf{B}}_n/Z({\mathsf{B}}_n)$. For cellular derivations satisfying a fixed-right-index compatibility condition, we construct $D$-connections on cell modules and give criteria for radical stability and descent to simple heads. We then study the non-semisimple regime $\delta =\kappa =0$, with $q$ not a root of unity. A general vector-space reduction of the outer derivation space is obtained, and in the fully degenerate rank-two case ${\mathsf{B}}_2(q,0,0)$, with $q+q^{-1}\ne 0$, the outer derivation Lie algebra is computed explicitly: it is two-dimensional and non-Abelian. Finally, using the known combinatorial bijection between the canonical bases of the relevant blob and Deguchi–Martin cell modules, we separate basis-level transport from algebra-level transport. We prove descent of Hecke-algebra derivations under stability of the defining quotient ideal; under an algebra isomorphism intertwining the module actions, $D$-connections transport by conjugation; and, under proportionality of the corresponding cellular Gram forms, radical stability is preserved. Low-rank examples illustrate the scope and limitations of these transfer statements.