What I am looking for is an abstract description of root data with their morphisms(!) plus a comparison with the categories of reductive groups over some field, Dynkin diagrams, Lie algebras, singularities...?

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    $\begingroup$ Btw, the nLab needs an article on this as well. $\endgroup$
    – Marc Palm
    Nov 21, 2012 at 9:31

1 Answer 1


The Isogeny Theorem does this for all isogenies between split reductive groups over any field (including non-central isogenies and exceptional isogenies in small characteristics) via the notion of "$p$-morphism" between root data ($p \ge 1$).

Is there a motivating situation to say more? For non-split groups the "anisotropic kernel" is invisible to root systems. For split semisimple groups in positive characteristic the homomorphisms into ${\rm{GL}}_n$ seem too rich to be encoded in terms of combinatorial information in a functorial way (e.g., there is a "highest weight" theory, but dimensions of irreducibles are mysterious, and semisimplicity fails badly). Complexities due to working over a ground field that isn't algebraically closed (even char. 0) seem huge, such as for field of definition of representations, much as we know very little about ${\rm{H}}^1(k,G)$ for general $k$ and many split $G$.

  • $\begingroup$ Which "Isogeny Theorem" is this? (It's not the Tate or Masser-Wüstholz one). Can you point me to some place to learn this stuff? $\endgroup$ Nov 21, 2012 at 15:43
  • $\begingroup$ @Konrad Voelkel: See Springer's book on algebraic groups and Steinberg's 1999 paper in Journal of Algebra over algebraically closed fields. General case follows by work with descent theory. $\endgroup$
    – user29283
    Nov 21, 2012 at 18:09

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