# Weyl Groups of Fine Gradings on Simple Lie Algebras of Types A, B, C and D

Elduque, Alberto; Kochetov, Mikhail

Serdica Mathematical Journal (2012)

- Volume: 38, Issue: 1-3, page 7-36
- ISSN: 1310-6600

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topElduque, Alberto, and Kochetov, Mikhail. "Weyl Groups of Fine Gradings on Simple Lie Algebras of Types A, B, C and D." Serdica Mathematical Journal 38.1-3 (2012): 7-36. <http://eudml.org/doc/288268>.

@article{Elduque2012,

abstract = {2010 Mathematics Subject Classification: Primary 17B70, secondary 17B40, 16W50.Given a grading Γ : L ⨁ = g ∈ G L g on a nonassociative algebra L by an abelian group G, we have two subgroups of Aut(L): the automorphisms that stabilize each component L g (as a subspace) and the automorphisms that permute the components. By the Weyl group of Γ 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 D 4) over an algebraically closed field of characteristic different from 2.∗ Supported by the Spanish Ministerio de Educación y Ciencia and FEDER (MTM 2010-18370-C04-02) and by the Diputación General de Aragón (Grupo de Investigación de Álgebra). ∗∗ Supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, Discovery Grant # 341792-07.},

author = {Elduque, Alberto, Kochetov, Mikhail},

journal = {Serdica Mathematical Journal},

keywords = {Graded Algebra; Fine Grading; Weyl Group; Simple Lie Algebra},

language = {eng},

number = {1-3},

pages = {7-36},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {Weyl Groups of Fine Gradings on Simple Lie Algebras of Types A, B, C and D},

url = {http://eudml.org/doc/288268},

volume = {38},

year = {2012},

}

TY - JOUR

AU - Elduque, Alberto

AU - Kochetov, Mikhail

TI - Weyl Groups of Fine Gradings on Simple Lie Algebras of Types A, B, C and D

JO - Serdica Mathematical Journal

PY - 2012

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 38

IS - 1-3

SP - 7

EP - 36

AB - 2010 Mathematics Subject Classification: Primary 17B70, secondary 17B40, 16W50.Given a grading Γ : L ⨁ = g ∈ G L g on a nonassociative algebra L by an abelian group G, we have two subgroups of Aut(L): the automorphisms that stabilize each component L g (as a subspace) and the automorphisms that permute the components. By the Weyl group of Γ 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 D 4) over an algebraically closed field of characteristic different from 2.∗ Supported by the Spanish Ministerio de Educación y Ciencia and FEDER (MTM 2010-18370-C04-02) and by the Diputación General de Aragón (Grupo de Investigación de Álgebra). ∗∗ Supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, Discovery Grant # 341792-07.

LA - eng

KW - Graded Algebra; Fine Grading; Weyl Group; Simple Lie Algebra

UR - http://eudml.org/doc/288268

ER -

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