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{Der}(\mathsf{B}_n) = \mathrm{Inn}(\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 $q+q^{-1}\neq0$, 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.