Weyl groups of fine gradings on simple Lie algebras of types A, B, C and D

Details Weyl groups of fine gradings on simple Lie algebras of types A, B, C and D Weyl groups of fine gradings on simple Lie algebras of types A, B, C and D

2011-09-16

arxiv.org/abs/1109.3540v1

Mathematics

Rings and Algebras

23 pages

Scientific paper

Given a grading on a nonassociative algebra by an abelian group, we have two subgroups of automorphisms attached to it: the automorphisms that stabilize each homogeneous component (as a subspace) and the automorphisms that permute the components. By the Weyl group of the grading we mean the quotient of the latter subgroup by the former. In the case of a Cartan decomposition of a semisimple complex Lie algebra, this is the automorphism group of the root system, i.e., the so-called extended Weyl group. A grading is called fine if it cannot be refined. We compute the Weyl groups of all fine gradings on simple Lie algebras of types A, B, C and D (except D4) over an algebraically closed field of characteristic different from 2.

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