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Weyl groups of fine gradings on simple Lie algebras of types \(A\), \(B\), \(C\) and \(D\). (English) Zbl 1359.17041

Given a grading \(\Gamma\): \(\mathcal L=\bigoplus_{g\in \Gamma}\mathcal L_g\) on a nonassociative algebra \(\mathcal L\) by an abelian group \(\Gamma\), one has the group of graded automorphisms \(\operatorname{Aut}({\mathcal L})\), i.e. such that the image of a graded component is another graded component. This group has a subgroup \(\text{Stab}(\mathcal L)\) consisting of automorphisms stabilizing each component \(\mathcal{L}_g\) (as a subspace). The authors refer to the quotient \(\operatorname{Aut}(\mathcal L)/\text{Stab}(\mathcal L)\) as the Weyl group of the grading. 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.
As a main result, the authors compute the Weyl groups of all fine gradings on simple Lie algebras of types \(A\), \(B\), \(C\) and \(D\) (except \(D_4\)) over an algebraically closed field of characteristic different from \(2\).
The situation with fine gradings on the simple Lie algebras of series \(A, B, C, D\) is complicated, because the fine gradings on matrix algebras yield only a part of the fine gradings on the simple Lie algebras of series \(A\) (so-called Type I gradings). In order to obtain the fine gradings for series \(B, C, D\) and the remaining (Type II) fine gradings for series \(A\), one has to consider so called fine \(\phi\)-gradings on matrix algebras, which were introduced and classified by A. Elduque [J. Algebra 324, No. 12, 3532–3571 (2010; Zbl 1213.17030)].
A classification, up to isomorphism, of the \(G\)-gradings on simple classical Lie algebras (different from \(D_4\)), for \(G\) a fixed abelian group, over an algebraically closed field of characteristic different from 2 is given in [Y. Bahturin and M. Kochetov, J. Algebra 324, No. 11, 2971–2989 (2010; Zbl 1229.17031)].

MSC:

17B70 Graded Lie (super)algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
16W50 Graded rings and modules (associative rings and algebras)
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