6
$\begingroup$

This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763. It got upvotes, but no answers or comments, and so I ask it here.

Let $G$ be a simply connected absolutely simple algebraic group of one of the types $^1{\sf A}_{n-1}$ (inner) or $^2{\sf A}_{n-1}$ (outer) over a field $k$. All such groups are described on page 55 of Tits, Classification of algebraic semisimple groups, Proc. Sympos. Pure Math. 9 (Boulder), 1966, pp. 33-61. The descriptions are as follows:

Type $^1{\sf A}_{n-1}$: Special linear group ${\rm SL}_m(D)$, where $D$ is a central division algebra of degree $d$ over $k$, and $n=md$.

Type $^2{\sf A}_{n-1}$: Special unitary group ${\rm SU}_m(D,h)$, where $D$ is a central division algebra of degree $d$ over a quadratic extension $K$ of $k$ with an involution of the second kind $\sigma\colon D\to D$ such that $k=\{x\in K\ |\ x^\sigma=x\}$, $$h\colon D^m\times D^m\to D$$ is a nondegenerate hermitian form relative to $\sigma$, and $n=md$.

Question. I am looking for a down-to-earth proof that all such groups are indeed of the form either ${\rm SL}_m(D)$ or ${\rm SU}_m(D,h)$. (For me, the Book of Involutions is not down-to-earth.)

I know that my group becomes ${\rm SL}_n(\bar k)$ over an algebraic closure $\bar k$ of $k$.

$\endgroup$
2
  • 3
    $\begingroup$ Is Section 24.f of Milne's Algebraic Groups not down-to-earth? $\endgroup$
    – anon
    Commented Oct 4, 2023 at 19:14
  • 1
    $\begingroup$ As usual, Milne's Algebraic Groups is very helpful. $\endgroup$ Commented Oct 5, 2023 at 15:22

1 Answer 1

6
$\begingroup$

Have you tried Chapter 17, section 17.1 from Springer's book on algebraic groups? I believe that is as down-to-earth as it can get, and it is certainly rather detailed.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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