# If split algebraic groups are potentially isomorphic, are they isomorphic?

Suppose $K$ is a (not necessarily algebraically closed) field, and $G_1$ and $G_2$ are split semisimple algebraic groups over $K$ which become isomorphic over $\bar{K}$, the algebraic closure of $K$. Are $G_1$ and $G_2$ isomorphic over $K$? What about if the $G$s are reductive?

It seems like this should follow (at least in the semisimple case) from Tits' general structure theorem for semisimple groups over a not-necessarily algebraically closed field; as explained in section 35.5 of Humphreys' Linear Algebraic Groups, a semisimple algebraic group is determined by its $\bar{K}$ isomorphism class, its anisotropic kernel (which looks like it is trivial for a split group) and its `index' (for which there again only seems to be one choice for a split group). But I am not expert enough to completely trust this argument...

• Does the existence of isotropic quadratic forms over Q which are not similar give a counterexample (via the orthogonal groups)? – Pete L. Clark Feb 15 '10 at 22:07
• Pete: I think these groups aren't going to be split in general. – Kevin Buzzard Feb 15 '10 at 23:16
• @Kevin: Yes, I see it now. I was thinking of the three-dimensional case for part of my argument and of the > three-dimensional case for another part. – Pete L. Clark Feb 15 '10 at 23:33
• If I am underanding things correctly now, the orthogonal group of a quadratic form will be split iff the quadratic form has maximal Witt index $\lfloor \operatorname{dim}(q)/2 \rfloor$? – Pete L. Clark Feb 16 '10 at 7:27
• @Pete: yes, that is correct. – George McNinch Apr 22 '10 at 16:40

[Caveat: that A.4 gives a complete treatment for the Isomorphism and Isogeny Theorems over general ground fields, and that is what the question is really about anyway; for the Existence Theorem in the case of exceptional types I don't know a way to "pull it down" from an algebraic closure, instead of having to revisit the constructions to make them work over prime fields or $\mathbf{Z}$.]
Here is another reference which seems very readable: A. Borel, J. Tits, Groupes réductifs Publ. Math. IHES , 27 (1965) pp. 55–150, Theorem 2.13: Two reductive $K$-split groups $G$ and $G'$ which are isomorphic over $\bar{K}$ are already isomorphic over $K$.