# Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$

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 $$k$$.

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