Tori_in_semisimple_groups Algebraic_torus

1 tori in semisimple groups

1.1 linear representations of tori
1.2 split rank of semisimple group
1.3 classification of semisimple groups

tori in semisimple groups
linear representations of tori

as seen in examples above tori can represented linear groups. alternative definition tori is:

a linear algebraic group torus if , if diagonalisable on algebraic closure.

the torus split on field if , if diagonalisable on field.

split rank of semisimple group

if




g



{\displaystyle \mathbf {g} }

semisimple algebraic group on field



f


{\displaystyle f}

then:

its rank (or absolute rank) rank of maximal torus subgroup in




g



{\displaystyle \mathbf {g} }

(note maximal tori conjugated on



f


{\displaystyle f}

rank well-defined);
its



f


{\displaystyle f}

-rank (sometimes called



f


{\displaystyle f}

-split rank) maximal rank of torus subgroup in



g


{\displaystyle g}

split on



f


{\displaystyle f}

.

obviously rank larger



f


{\displaystyle f}

-rank; group called split if , if equality holds (that is, there maximal torus in




g



{\displaystyle \mathbf {g} }

split on



f


{\displaystyle f}

). group called anisotropic if contains no split tori (i.e.



f


{\displaystyle f}

-rank zero).

classification of semisimple groups

in classical theory of semisimple lie algebras on complex field cartan subalgebras play fundamental rôle in classification via root systems , dynkin diagrams. classification equivalent of connected algebraic groups on complex field, , cartan subalgebras correspond maximal tori in these. in fact classification carries on case of arbitrary base field under assumption there exists split maximal torus (which automatically satisfied on algebraically closed field). without splitness assumption things become more complicated , more detailed theory has developed, still based in part on study of adjoint actions of tori.

if




t



{\displaystyle \mathbf {t} }

maximal torus in semisimple algebraic group




g



{\displaystyle \mathbf {g} }

on algebraic closure gives rise root system



Φ


{\displaystyle \phi }

in vector space



v
=

x

∗


(

t

)

⊗


z




r



{\displaystyle v=x^{*}(\mathbf {t} )\otimes _{\mathbb {z} }\mathbb {r} }

. on other hand, if








f



t

⊂

t



{\displaystyle {}_{f}\mathbf {t} \subset \mathbf {t} }

maximal



f


{\displaystyle f}

-split torus action on



f


{\displaystyle f}

-lie algebra of




g



{\displaystyle \mathbf {g} }

gives rise root system








f


Φ


{\displaystyle {}_{f}\phi }

. restriction map




x

∗


(

t

)
→

x

∗



(

f



t

)


{\displaystyle x^{*}(\mathbf {t} )\to x^{*}(_{f}\mathbf {t} )}

induces map



Φ
→





f


Φ
∪
{
0
}


{\displaystyle \phi \to {}_{f}\phi \cup \{0\}}

, tits index way encode properties of map , of action of galois group of





f
¯



/

f


{\displaystyle {\overline {f}}/f}

on



Φ


{\displaystyle \phi }

. tits index relative version of absolute dynkin diagram associated



Φ


{\displaystyle \phi }

; obviously, finitely many tits indices can correspond given dynkin diagram.

another invariant associated split torus








f



t



{\displaystyle {}_{f}\mathbf {t} }

anisotropic kernel: semisimple algebraic group obtained derived subgroup of centraliser of








f



t



{\displaystyle {}_{f}\mathbf {t} }

in




g



{\displaystyle \mathbf {g} }

(the latter reductive group). name indicates anisotropic group, , absolute type uniquely determined








f


Φ


{\displaystyle {}_{f}\phi }

.

the first step towards classification following theorem

two semisimple



f


{\displaystyle f}

-algebraic groups isomorphic if , if have same tits indices , isomorphic anisotropic kernels.

this reduces classification problem anisotropic groups, , determining tits indices can occur given dynkin diagram. latter problem has been solved in tits (1966). former related galois cohomology groups of



f


{\displaystyle f}

. more precisely each tits index there associated unique quasi-split group on



f


{\displaystyle f}

; every



f


{\displaystyle f}

-group same index inner form of quasi-split group, , classified galois cohomology of



f


{\displaystyle f}

coefficients in adjoint group.