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A tower of algebras is a sequence of algebras $$A_0 \hookrightarrow A_1 \hookrightarrow \cdots \hookrightarrow A_n \hookrightarrow \cdots$$ with embeddings $A_n \otimes A_m \hookrightarrow A_{n+m}$ satisfying an associativity condition.

I found this definition recently in papers in representation theory such as:

"Algebraic structures on Grothendieck groups of a tower of algebras" by Huilan Li and Nantel Bergeron

"Combinatorial Hopf algebras and Towers of Algebras - Dimension, Quantization, and Functoriality" by Nantel Bergeron, Thomas Lam and Huilan Li

"Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras" by F. Hivert and N. Thiéry

Also, I found this definition in a homotopy theoretic (and operadic) paper: "On Quillen homology and a homotopy completion tower for algebras over operads" by J. Harper and K. Hess.

My question is: where does the definition of tower of algebras come from and what was the original motivation to introduce this object?

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up vote 1 down vote accepted

I'm a bit out of my comfort zone with this, but Goodman, de la Harpe and Jones "Coxeter graphs and towers of algebras" might be a place to start. The work here was motivated by questions about von Neumann algebras.

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I don't know what the original motivation was (or where the original definition was). But one reasonable conceptual explanation for the definition is this: let $\mathcal C$ be a category with objects $\mathbb N$, endomorphisms $A_n$, and other morphism spaces 0. Then the embeddings $A_m \otimes A_n \to A_{m+n}$ (and the associativity conditions) are what is needed to provide $\mathcal C$ with a strict monoidal structure, where $m \otimes n := m + n$. Then one reason towers of algebras tend to be interesting is that monoidal categories tend to be interesting.

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In particular, 1) the group algebras of the symmetric groups arise as a tower in this way with $C$ the free symmetric monoidal $k$-linear category on an object, 2) the group algebras of the braid groups arise as a tower in this way with $C$ the free braided monoidal $k$-linear category on an object, and 3) the Temperley-Lieb algebras arise as a tower in this way with $C$ the free monoidal $k$-linear category on a self-dual object. – Qiaochu Yuan Jun 1 at 3:06

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