Mathematics of the USSR-Izvestiya · 1976 · 249 citations · 11 references
Natural SciencesMathematicsAlgebraGeometryRepresentation TheoryAbstract AlgebraAlgebraic GeometryGroup TheoryHomological AlgebraLie AlgebrasLie GroupsGraded Lie AlgebrasGroup SubgroupsLie Algebra Automorphism GroupsLie Algebra ModulesLie Algebra TheoryReductive Group InvariantsReductive Lie GroupsWeyl Groups
The study examines the action of the fixed‑point subgroup G₀ of a semisimple automorphism θ on an eigenspace V within the Lie algebra 𝔤 of a reductive algebraic group G. The authors introduce θ‑groups, the linear groups arising from this action, and aim to investigate their structural properties analogous to those of adjoint groups. They define Cartan subgroups and Weyl groups for θ‑groups, establishing the framework for further analysis. They prove that the Weyl group of a θ‑group is generated by complex reflections, implying that its invariant algebra is freely generated. The paper includes a bibliography of 30 references.
The action of the group G0 of fixed points of a semisimple automorphism θ of a reductive algebraic group G on an eigenspace V of this automorphism in the Lie algebra g of the group G is considered. The linear groups which are obtained in this manner are called θ-groups in this paper; they have certain properties which are analogous to properties of the adjoint group. In particular, the notions of Cartan subgroup and Weyl group can be introduced for θ-groups. It is shown that the Weyl group is generated by complex reflections; from this it follows that the algebra of invariants of any θ-group is free. Bibliography: 30 titles.
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Finite Unitary Reflection Groups
G. C. Shephard, John A. Todd · Canadian Journal of Mathematics · 1954 · 1.1K citations · Full text
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