# Diagram 16: Brauer algebra

This structure is named after **Richard Dagobert Brauer**, who first published about it in 1937. He's also known for a theorem by Brauer and **K. A. Fowler** which has been important for the classification of finite simple groups. He's contributed to number theory, invented modular representation theory, and more - so he's quite a star!

I got the photo from the wikipedia page about him. The other person is his wife **Ilse Brauer**, also a mathematician. Reading about researchers of that time is reading about exciting times for mathematics. You could as well start right here:

http://en.wikipedia.org/wiki/Richard_Brauer

How does one work in a *Brauer algebra*? Let me explain: The objects of this algebra are graphs, constructed over a pair of sets of n elements (call them x_i and y_i). To get a graph from this pecularily labeled set we have to add edges - like this:

Each element is paired with one other element (is connected by one edge). That is, we divide the elements into couples (form a *complete matching*). If we only allow to pair y's with x's we get a *representation* for permutations. But here, we're allowed to connect horizontally as well: An x_i may be linked to some x_j or an y_i to some y_j (then, of course, i must not be equal to j!).

We still need a rule to compose two such graphs A and B to obtain a new graph C = A * B. Here it is: Glue A on top of B, identifying A_y_i with B_x_i. Now connect A_x_i with B_y_j when those are connected by a path.

Note that loops may occur, but they aren't important for the result and may be discarded. If it's not clear yet, look at the diagram below - taken from the arxiv paper linked further down, or try the picture wikipedia shows here:

http://en.wikipedia.org/wiki/Brauer_algebra

Unfortunately, i could not find a free version of Brauer's original 1937 paper *'On Algebras Which are Connected with the Semisimple Continuous Groups'*. The title suggests he was working on Lie groups when discovering his algebra... But simple things first:

The permutation graphs (without horizontal edges) form a subset of all Brauer graphs, and composing two permutation graphs yields another permutation graph. That means that permutations "sit" comfortably inside Brauer algebras (they form a subalgebra ). Our algebra can't be called a group because not all elements are invertible .

Brauer's algebra can be used to construct a representation for the orthogonal group O_n (rotations in dimension n)! The orthogonal group O_n is a subgroup of the general linear group GL_n (the group of all size n square matrices, over some field ).

One can use the permutations (of n elements, also known as the symmetric group S_n) to construct a representation for the general linear group GL_n. The Brauer algebra does the same job for the orthogonal group O_n.

But that's curious. Why? Stare hard:

While S_n => GL_n (S_n represents a basis for GL_n), and B_n => O_n (B_n represents a basis for O_n), and further S_n c B_n (S_n is a subalgebra of B_n), but: O_n c GL_n (O_n is a subgroup of GL_n).

I'm not sure what to make of that, i probably should learn more representation theory ...

Here's a paper "the blocks of the Brauer algebra in characteristic zero" by Austin Cox , Maud de Visscher and Paul Martin (2007). It served me well as an exposition on the application of Brauer algebras . They talk about the symplectic group Sp_n, which is the complex sibling of O_n. Interestingly, they compare the theory of B_n(C) with theories of other, well known algebras:

http://arxiv.org/abs/math/0601387

One notable genralization is given by the Birmanâ€“Wenzl algebra (also known as BMW algebra ). It's a "deformation" of the Brauer algebra and isomorphic to +Louis Kauffman 's tangle algebra . BMW contains the braid group and all of these are topics for another time.

http://en.wikipedia.org/wiki/Birman-Wenzl_algebra

http://en.wikipedia.org/wiki/Braid_group

Below, you see a Bratelli diagram (Update: Erm. No you don't. I forgot. Sorry, next time...) depicting Brauer algebras and BMW algebras on 0 to 4 strands. A Bratelli diagram can actually display a lot of other algebras, too, and i'll tell you about it... at a later time. Stay tuned!