@article{Suhail2026, 
author = {Muntasir Suhail and Osman Abdalla Adam Osman and Mohammed Rabih and Mohammad Shane Alam},
title = {Symmetries of blob algebras: Derivations, outer classes and cellular transfer},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {7},
pages = {22410-22431},
keywords = {blob algebra, derivation, cellular algebra, Temperley–Lieb algebra, Hochschild cohomology, outer derivation, Deguchi–Martin quotient, mathematical operators},
url = {https://www.sciopen.com/article/10.3934/math.2026907},
doi = {10.3934/math.2026907},
abstract = {We study derivations of the blob algebra              B        n    (  q  ,  δ  ,  κ  ) 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        D    e    r    (            B        n    )  =      I    n    n    (            B        n    )  ≅            B        n        /    Z  (            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    δ  =  κ  =  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              B        2    (  q  ,  0  ,  0  ), with    q  +      q          −      1        ≠  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.}
}