Premium
Weyl Groups are Finite — and Other Finiteness Properties of Cartan Subalgebras
Author(s) -
Hofmann Karl H.,
Lawson Jimmie D.,
Ruppert Wolfgang A. F.
Publication year - 1996
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19961790109
Subject(s) - mathematics , complexification , cartan subalgebra , subalgebra , pure mathematics , centralizer and normalizer , weyl group , lie algebra , cartan matrix , representation theory , kernel (algebra) , automorphism , adjoint representation , algebra over a field , lie conformal algebra , adjoint representation of a lie algebra
For each pair (,) consisting of a real Lie algebra and a subalgebra a of some Cartan subalgebra of such that [, ]∪ [, ] we define a Weyl group W(, ) and show that it is finite. In particular, W(, ,) is finite for any Cartan subalgebra h. The proof involves the embedding of 0 into the Lie algebra of a complex algebraic linear Lie group to which the structure theory of Lie algebras and algebraic groups is applied. If G is a real connected Lie group with Lie algebra , the normalizer N (, G) acts on the finite set Λ of roots of the complexification c with respect to hc, giving a representation π : N (, G )→ S(Λ) into the symmetric group on the set Λ. We call the kernel of this map the Cartan subgroup C() of G with respect to h; the image is isomorphic to W(, ), and C ()= { g G : Ad( g )( h )— h ε [h,h] for all h ε h }. All concepts introduced and discussed reduce in special situations to the familiar ones. The information on the finiteness of the Weyl groups is applied to show that under very general circumstance, for b ∪ the set ⊂ ϕ(b) remains finite as ϕ ranges through the full group of inner automorphisms of .