Robert's Mistakes

Diagram 16: Brauer algebra

Feb 06, 2014

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:

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:

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:

One notable genralization is given by the Birman–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.

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!