Gradings on Finite-Dimensional Simple Lie Algebras

Kochetov, Mikhail

doi:10.1007/s10440-008-9386-0

Cited by 53 publications

(59 citation statements)

References 36 publications

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“…This group G has the following property: given any group grading A = h∈H A h for an abelian group H which is a coarsening of Γ , then there exists a unique homomorphism f : G → H such that A h = g∈ f −1 (h) A g (see [Koch09]). The group G is called the universal grading group of Γ .…”

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confidence: 98%

“…Actually, any group grading on a finite dimensional simple Lie algebra is an abelian group grading (see [Koch09,Proposition 3], where a very complete survey of many results and references on gradings on Lie algebras can be found). Hence a fine grading will refer to an abelian group grading which admits no proper refinement in the class of abelian group gradings.…”

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confidence: 99%

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Fine gradings on simple classical Lie algebras

Elduque

2010

Journal of Algebra

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Fine gradings on simple classical Lie algebras

Elduque

2010

Journal of Algebra

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“…In fact, one may expect that there are problems which are naturally and more simply formulated exploiting bases dictated by Σ-multiplicative and of maximal length gradings. Let us illustrate this fact by considering the particular case of the (associative) matrix algebra M n (K), in which the gradings induced by the Pauli matrices have been fundamental in the study of the mathematical physics problems in [16] and [19], (see also [25]), as consequence of being Σ-multiplicative and of maximal length. The general reason for that is the fact that such gradings decompose M n (K) into the direct sum of n 2 subspaces of dimension 1 (i.e.…”

Section: Finally Proposition 31-2 Gives Usmentioning

confidence: 99%

On graded matrix Hom-algebras

Periñán¹,

Martín²

2012

ELA

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Abstract. Consider an (associative) matrix algebra M I (R) graded by means of an abelian group G, and a graded automorphism φ on M I (R). By defining a new product by x ⋆ y := φ(x)φ(y) on M I (R), (M I (R), ⋆) becomes a hom-associative algebra graded by a twist of G. The structure of (M I (R), ⋆) is studied, by showing that M I (R) is of the formwith U an R-submodule of the 0-homogeneous component and any I j a well described graded ideal of M I (R), satisfying I j ⋆ I k = 0 if j = k. Under certain conditions, the graded simplicity of an arbitrary graded hom-associative algebra M is characterized and it is shown that M is the direct sum of the family of its simple graded ideals.

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“…[Koc09] or [EK13, Proposition 1.12]). From now on, we will assume that all gradings are by abelian groups, which will be written additively.…”

Section: Introductionmentioning

confidence: 99%

Counting Fine Gradings on Matrix Algebras and on Classical Simple Lie Algebras

Kochetov

Parsons

Sadov

2013

Int. J. Algebra Comput.

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Abstract. Known classification results allow us to find the number of (equivalence classes of) fine gradings on matrix algebras and on classical simple Lie algebras over an algebraically closed field F (assuming char F = 2 in the Lie case). The computation is easy for matrix algebras and especially for simple Lie algebras of type Br (the answer is just r + 1), but involves counting orbits of certain finite groups in the case of Series A, C and D. For X ∈ {A, C, D}, we determine the exact number of fine gradings, N X (r), on the simple Lie algebras of type Xr with r ≤ 100 as well as the asymptotic behaviour of the average,N X (r), for large r. In particular, we prove that there exist positive constants b and c such that exp(br 2/3 ) ≤N X (r) ≤ exp(cr 2/3 ). The analogous average for matrix algebras Mn(F) is proved to be a ln n + O(1) where a is an explicit constant depending on char F.

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Gradings on Finite-Dimensional Simple Lie Algebras

Cited by 53 publications

References 36 publications

Fine gradings on simple classical Lie algebras

Fine gradings on simple classical Lie algebras

On graded matrix Hom-algebras

Counting Fine Gradings on Matrix Algebras and on Classical Simple Lie Algebras

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