We present a uniform description of the center of the Temperley–Lieb algebra \(TL_n(\delta)\) for all nonzero complex parameters \(\delta\). In particular, we show that the center has dimension \[dim(Z(TL_n(\delta)) = 1 +\lfloor\frac{n}{2}\rfloor\] for every\(n\) and \(\delta\), and we describe a basis arising from a natural filtration closely tied to the cellular structure of \(TL_n(\delta)\). Our approach makes use of the diagrammatic and algebraic presentation of Temperley–Lieb algebras, their representation theory, and deformation theoretic methods.

In studying Hopf algebras \(H=\bigoplus_{n\geq0} \mathbb{Z}P_n\) built on combinatorial gadgets \(P_\bullet\) with a natural poset structure, it is natural (and often fruitful) to use the poset to define new bases for \(H\). Here I introduce a variation on the common technique in the presence of poset maps \(f_n:P_n \to Q_n\). After introducing the basic framework, I'll give several examples and a couple consequences. As the title suggests, the former requires the maps \(f_n\) to be part of an *adjoint modality. Based on joint work with Marcelo Aguiar. (In progress.).

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Here is where the abstract will appear.

Here is where the abstract will appear.

Here is where the abstract will appear.

Here is where the abstract will appear.