# Reference for quaternions and complexification

I am writing something on the complexification of a real associative algebra. There are two well-known isomorphisms about quaternions: $$\mathbb{H}\otimes_\mathbb{R}\mathbb{C}\simeq M_2(\mathbb{C})$$ and $$\mathbb{H}\otimes_\mathbb{R}\mathbb{H}\simeq M_4(\mathbb{R}).$$ I need a reference which includes both of these two isomorphisms. I also wonder if there is a reference dealing with the bimodules over the real division rings and their complexification. Anything about complexification and related Galois descent are also welcome! Thanks a lot!

## 1 Answer

What you are looking for is contained in

T. Y. Lam: Introduction to quadratic forms over fields, Graduate Studies in Mathematics 67. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-1095-2/hbk). xxi, 550 p. (2005). ZBL1068.11023.

In particular:

• the isomorphism $$\mathbb{H}\otimes_\mathbb{R}\mathbb{C}\simeq M_2(\mathbb{C})$$ is a consequence of the proof of parts (3), (4) of Proposition 1.1 (taking, in the Author's notation, $$F=\mathbb{R}$$, $$E=\mathbb{C}$$, $$a=b=-1$$), see the beginning of p. 53;
• the isomorphism $$\mathbb{H}\otimes_\mathbb{R}\mathbb{H}\simeq M_4(\mathbb{R})$$ is the content of Exercise 9 p. 76, with $$F=\mathbb{R}$$.
• Thanks! This helps a lot. Do you know something which is more specialized on real algebras and complexification? – yzl Nov 2 at 3:43
• You are welcome. At the moment, I have no other reference in mind... – Francesco Polizzi Nov 2 at 9:12