Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am currently reading the book Spin Geometry by Lawson/Michelsohn to understand Dirac Operators and related topics. At some point it uses representation theory to classify Clifford Algebras. In particular this book states in I,§5. Theorem 5.6 on p.31:

Let $K=\mathbb{R}, \mathbb{C}$ or $\mathbb{H}$ and consider the ring $K(n)$ of $n \times n$ $K$-matrices as an algebra over $\mathbb{R}$. Then the natural representation $\rho$ of $K(n)$ on the vector space $K^n$ is, up to equivalence, the only irreducible representation of $K(n)$. The algebra $K(n) \oplus K(n)$ has exactly two equivalence classes of irreducible representations. They are given by $$ \rho_1(\varphi_1,\varphi_2) = \rho(\varphi_1) \text{ and } \rho_2(\varphi_1,\varphi_2) =\rho(\varphi_2) $$ acting on $K(n)$.

I am perfectly fine with the statement, but I would like to see a proof of that. Lawson/Michelsohn claim that this follows from the fact that the algebras $K(n)$ are simple and that simple algebras have only one irreducible representation up to equivalence. For the details the reader is referred to Lang: Algebra. Since this book as nearly 1000 pages I would like to ask, if someone could be a little more precise? Of course I would also be happy with a direct proof of the claim or a reference to any other readably textbook containing a proof. I must point out that I have no prior experience in the field of representation theory.

share|improve this question
    
Wait a sec, it's not clear from your question: do you want a reference to something other than Lang? Even if so, I encourage you to read just the section of Lang containing the theorems mentioned below; you will not find a more self-contained treatment. –  S123 Mar 7 '12 at 1:50
    
As I said, I would be happy with either a direct proof, a more precise reference to Lang or a precise reference to another textbook. –  Meneldur Mar 8 '12 at 16:21
add comment

1 Answer 1

up vote 2 down vote accepted

The specific reference: look at Theorems 4.3 and 4.4 on page 653 of the revised third edition (does that make it edition 3.5?) of Lang's "Algebra". These give you precisely the statements you want. The proofs are pretty readable, in my opinion, but if you like I will give more details here.

To apply these theorems to your situation, observe that each of the matrix algebras you are looking at is the direct sum of $n$ left ideals, the $i$th of which consists of matrices which have non-zero entries only in the $i$th column. Each of these ideals is simple, and they are all isomorphic as left modules to $K^n$. This is the definition of simple ring given by Lang: it is a semisimple ring (that is, is a direct sum of simple ideals) with only one isoclass of left ideals appearing in the direct sum decomposition. This also shows that the direct sum of two copies of your matrix algebra is semisimple, and Lang's Thm 4.4 gives the statement about its representations that you need.

share|improve this answer
    
Thanks, now I see much better, why Lang is helpful here. –  Meneldur Mar 8 '12 at 16:22
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.