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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 http://arxiv.org/pdf/math/0612170

"Combinatorial Hopf algebras and Towers of Algebras - Dimension, Quantization, and Functoriality" by Nantel Bergeron, Thomas Lam and Huilan Li http://arxiv.org/pdf/0903.1381

"Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras" by F. Hivert and N. Thiéry http://garsia.math.yorku.ca/fpsac06/papers/75.pdf

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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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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    $\begingroup$ 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. $\endgroup$
    – Qiaochu Yuan
    Commented Jun 1, 2016 at 3:06
  • $\begingroup$ Well, thank your setting and @QiaochuYuan examples (to which we could add deformations as the Hecke algebras, degenerate - no group, but a monoid there - or not :). Are there good references where your categorical-skeletic ("$\mathcal C$ be a category with objects $\mathbb N$") is introduced and elaborated ? Thanks in advance. $\endgroup$
    – Duchamp Gérard H. E.
    Commented Jul 3, 2021 at 7:23
  • $\begingroup$ Am I right to say that it is only with the "external" multiplication $A_m \otimes A_n \rightarrow A_{m+n}$ (presumably this is horizontal concatenation of diagrams in the case of a tower of planar algebras) that the graded vector space $H := \bigotimes_{n=0}^\infty A_n/[A_n,A_n]$ acquires the structure of an associative, unital algebra; i.e. the "internal" multiplication coming from the inclusion maps $A_m \subset A_n \subset A_{m+n}$ (in the planar algebra case this would be mating diagrams vertically) does not, in general, make $H$ into an algebra ? $\endgroup$
    – Jeanne Scott
    Commented Jun 18, 2022 at 14:37
  • $\begingroup$ I would guess that in general the maps $A_m\otimes A_n \to A_{m+n}$ are strictly more data than the inclusions $A_n \subset A_{n+1}$. As far as I know you need the first maps to make $H$ an algebra. $\endgroup$
    – Peter Samuelson
    Commented Jun 19, 2022 at 19:27

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