Group Gradings on Lie Algebras, with Applications to Geometry. I

Developments and Retrospectives in Lie Theory - Developments in Mathematics ◽

10.1007/978-3-319-09804-3_1 ◽

2014 ◽

pp. 1-50 ◽

Cited By ~ 2

Author(s):

Yuri Bahturin ◽

Michel Goze ◽

Elisabeth Remm

Keyword(s):

Lie Algebras ◽

Group Gradings

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GROUP GRADINGS ON FINITARY SIMPLE LIE ALGEBRAS

International Journal of Algebra and Computation ◽

10.1142/s0218196712500464 ◽

2012 ◽

Vol 22 (05) ◽

pp. 1250046 ◽

Cited By ~ 3

Author(s):

YURI BAHTURIN ◽

MATEJ BREŠAR ◽

MIKHAIL KOCHETOV

Keyword(s):

Abelian Group ◽

Lie Algebras ◽

Vector Spaces ◽

Closed Field ◽

Dimensional Vector ◽

Linear Transformations ◽

Special Linear ◽

Infinite Dimensional ◽

Group Gradings ◽

Arbitrary Abelian Group

We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.

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Arithmetically-free group-gradings of Lie algebras: II

Journal of Algebra ◽

10.1016/j.jalgebra.2017.09.014 ◽

2017 ◽

Vol 492 ◽

pp. 457-474 ◽

Cited By ~ 1

Author(s):

Wolfgang Alexander Moens

Keyword(s):

Lie Algebras ◽

Free Group ◽

Group Gradings

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Group gradings on restricted Cartan-type Lie algebras

Pacific Journal of Mathematics ◽

10.2140/pjm.2011.253.289 ◽

2011 ◽

Vol 253 (2) ◽

pp. 289-319

Author(s):

Yuri Bahturin ◽

Mikhail Kochetov

Keyword(s):

Lie Algebras ◽

Cartan Type ◽

Group Gradings

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Group Gradings on Finite Dimensional Lie Algebras

Algebra Colloquium ◽

10.1142/s1005386713000540 ◽

2013 ◽

Vol 20 (04) ◽

pp. 573-578 ◽

Cited By ~ 4

Author(s):

Dušan Pagon ◽

Dušan Repovš ◽

Mikhail Zaicev

Keyword(s):

Finite Group ◽

Lie Algebras ◽

Characteristic Zero ◽

Closed Field ◽

Algebraically Closed Field ◽

Solvable Radical ◽

Finite Dimensional ◽

Levi Subalgebra ◽

Group Gradings ◽

Finite Dimensional Lie Algebras

We study gradings by non-commutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if L is graded by a non-abelian finite group G, then the solvable radical R of L is G-graded and there exists a Levi subalgebra B=H1⊕ ⋯ ⊕ Hm homogeneous in G-grading with graded simple summands H1,…,Hm. All Supp Hi (i=1,…,m) are commutative subsets of G.

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Group Gradings on Filiform Lie Algebras

Communications in Algebra ◽

10.1080/00927872.2014.937535 ◽

2015 ◽

Vol 44 (1) ◽

pp. 40-62 ◽

Cited By ~ 6

Author(s):

Yuri Bahturin ◽

Michel Goze ◽

Elisabeth Remm

Keyword(s):

Lie Algebras ◽

Group Gradings ◽

Filiform Lie Algebras

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Multiplicity-Free Gradings on Semisimple Lie and Jordan Algebras and Skew Root Systems

Algebra Colloquium ◽

10.1142/s1005386719000129 ◽

2019 ◽

Vol 26 (01) ◽

pp. 123-138

Author(s):

Gang Han ◽

Yucheng Liu ◽

Kang Lu

Keyword(s):

Abelian Group ◽

Lie Algebras ◽

Root Systems ◽

Jordan Algebras ◽

Homogeneous Component ◽

New Approach ◽

Jordan Type ◽

Semisimple Lie Algebras ◽

Group Gradings ◽

Multiplicity Free

A G-grading on an algebra, where G is an abelian group, is called multiplicity-free if each homogeneous component of the grading is 1-dimensional. We introduce skew root systems of Lie type and skew root systems of Jordan type, and use them to construct multiplicity-free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp., Jordan) algebras are simple. Two families of skew root systems of Lie type (resp., of Jordan type) are constructed and the corresponding Lie (resp., Jordan) algebras are identified. This is a new approach to study abelian group gradings on Lie and Jordan algebras.

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