Welcome to MathOverflow

Question

4

1

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?

edited Sep 19 2011 at 8:42

asked Sep 18 2011 at 20:31

Yannic
195●1●2●5

Chris Godsil · Accepted Answer · 2011-09-19 13:44:35Z UTC

1

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.

answered Sep 19 2011 at 13:44

Chris Godsil
5,694●1●8●22

Where does the definition of “Tower of Algebras” come from?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.