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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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