Are certain simple Lie groups linear algebraic groups?

2012-09-19T15:01:08Z

Assume you have an almost connected simple Lie group G with trivial center. (In particular excluding non-algebraic examples such as the universal cover of SL_2(R).)

Such a group should automatically be an algebraic group over the reals resp. the complex numbers.

Is this true and why?

Can we in addition conclude (EDIT: under a good choice of the field and possibly additional assumptions?) that G is absolutely almost simple as an algebraic group?

EDIT: Asking this I do not want to regard a complex Lie group as a real algebraic group.

Answer by Vesselin Dimitrov for Are certain simple Lie groups linear algebraic groups?

2012-09-19T15:56:13Z

The answer is yes for complex Lie groups, and follows from the classification. (Root data are in fact defined over $\mathbb{Z}$: a complex semisimple group has not only an underlying algebraic, but even an arithmetic structure). For a more direct explanation, see Theorem 6.3 in the book "Lie Groups and Lie Algebras III" by Onischik-Vinberg: any connected complex Lie group satisfying $G = [G,G]$ and admitting a faithful linear representation (which for semisimple groups is automatic), has a unique underlying complex algebraic structure.

For real Lie groups this is not quite true, as noted above.

Are certain simple Lie groups linear algebraic groups? - MathOverflow

Are certain simple Lie groups linear algebraic groups?

Answer by Vesselin Dimitrov for Are certain simple Lie groups linear algebraic groups?