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?
> 
> 
> 



Where does the definition of "Tower of Algebras" come from?
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.


