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Draper, Cristina; Martín, Cándido

doi:10.1016/j.laa.2006.01.017

Cited by 44 publications

(20 citation statements)

References 11 publications

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“…real form of sl(4, C) real form of sp(4, C) real form of o(4, C) sl(4, R) sp(4, R) su * (4) so * (4) su (4,0) usp(4, 0) so(4, 0) su (3,1) so(3, 1) su(2, 2) usp(2, 2) so(2, 2) Table 25. The full list of the real forms of sp(4, C) and of o(4, C), and their relation to the real forms of sl(4, C).…”

Section: Fine Group Gradings Of the Realmentioning

confidence: 99%

“…• The real form so(3, 1) has fine group gradings determined by Γ 3 , Γ 4 , Γ 8 : Γ 3 → so(3, 1) = o K1 (4, C) ∩ su E1 (3, 1) Γ 4 , Γ 8 → so(3, 1) = o K2 (4, C) ∩ su E2 (3, 1) • The real form so(2, 2) has fine group gradings determined by Γ 3 , Γ 4 , Γ 5 , Γ 6 , Γ 7 , and Γ 8 : Γ 3 , Γ 7 → so(2, 2) = o K1 (4, C) ∩ su E1 (2, 2) Γ 4 → so(2, 2) = o K2 (4, C) ∩ su E2 (2, 2) Γ 5 , Γ 6 → so(2, 2) = o K3 (4, C) ∩ su E3 (2, 2) Γ 8 → so(2, 2) = o K2 (4, C) ∩ su E5 (2, 2) Table 27. Summary of fine group gradings of the real forms of o(4, C) displayed by the fine group gradings of the real forms of sl(4, C).…”

Section: Fine Group Gradings Of the Real Forms Of O(4 C)mentioning

confidence: 99%

“…References [14,15,16] are the departure points of the present work. Gradings of the exceptional Lie algebra g 2 are listed in [4], gradings of f 4 can be found in [5]. …”

mentioning

confidence: 99%

“…If a refinement of a group gradings is also a group grading, then we call it group refinement. There is a relationship between the group labelling the group refinement and the group labelling the original group grading (see [4]). More precisely, the index set of a group grading can be embedded into a universal Abelian group, say G, such that the index set of any coarsening is embedable into an image of G by a group epimorphism.…”

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confidence: 99%

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Fine Group Gradings of the Real Forms of sl(4,C), sp(4,C), and o(4,C)

Patera

Pelantová

Svobodová

2007

Journal of Mathematical Physics

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Abstract. We present an explicit description of the 'fine group gradings' (i.e. group gradings which cannot be further refined) of the real forms of the semisimple Lie algebras sl(4, C), sp(4, C), and o(4, C). All together 12 real Lie algebras are considered, and the total of 44 of their fine group gradings are listed.The inclusions sl(4, C) ⊃ sp(4, C) ⊃ o(4, C) are an important tool in our presentation. Systematic use is made of the faithful representations of the three Lie algebras by 4 × 4 matrices. IntroductionWide diversity of applications of low dimensional semisimple Lie algebras, real or complex, is witnessed by the innumerable papers found in the literature, where such Lie algebras play a role. In the article we consider the real forms of the three semisimple Lie algebras which are faithfully represented by 4 × 4 matrices. More precisely, we have the Lie algebras and the inclusions among them:sl(4, C) ⊃ sp(4, C) ⊃ o(4, C).Here sl(4, C) is the Lie algebra of all traceless matrices C 4×4 . Then sp(4, C) is the Lie algebra of all (symplectic) transformations which preserve a skew-symmetric bilinear form in the 4-space C 4 . Finally, o(4, C) is the Lie algebra of all (orthogonal) transformations which preserve a symmetric bilinear form in C 4 . The second inclusion is somewhat misleading and therefore deserves a comment. Due to the fact that o(4, C) ≃ sl(2, C) × sl(2, C) is not simple, one has a choice considering o(4, C) as linear transformations in C 4 . One can introduce symmetric or skew-symmetric bilinear form in C 4 invariant under the matrices of o(4, C). In the symmetric case, o(4, C) contains all such transformations, while all symplectic transformations are in sp(4, C) and only some of them in o(4, C).The list of the real forms considered in this paper is the following:sl(4, C) : sl(4, R), su * (4), su(4, 0), su(3, 1), su(2, 2)sp(4, C) : sp(4, R), usp(4, 0), usp(2, 2)o(4, C) : so * (4), so(4, 0), so(3, 1), so(2, 2)The subject of our paper are the gradings of these real forms. For a general motivation for studying the gradings, we can point out [10,11,12,13,20,21]. Applications of the real forms are too numerous to be mentioned here. For example, let us just mention that among the orthogonal realizations of B 2 there are the Lie algebras of the de Sitter groups, that some real forms of the symplectic realization have been intensively applied in the recent years in nuclear physics, and also in quantum optics [18,19,1,3].A grading carries basic structural information about its Lie algebra. That is particularly true about the fine grading. One type of a grading is obtained by decomposing the Lie algebra into eigensubspaces of mutually commuting automorphisms. This grading has an interesting property: Its grading subspaces can be indexed by elements of an Abelian group. Such gradings are called group gradings. The most famous example of a grading of this type, obtained by means of automorphisms from the MAD-group which is the maximal torus of the corresponding Lie group, is called the root decompositio...

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Section: Fine Group Gradings Of the Realmentioning

confidence: 99%

Section: Fine Group Gradings Of the Real Forms Of O(4 C)mentioning

confidence: 99%

“…References [14,15,16] are the departure points of the present work. Gradings of the exceptional Lie algebra g 2 are listed in [4], gradings of f 4 can be found in [5]. …”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Fine Group Gradings of the Real Forms of sl(4,C), sp(4,C), and o(4,C)

Patera

Pelantová

Svobodová

2007

Journal of Mathematical Physics

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“…For the classical simple Lie algebra over an algebraically closed field of characteristic 0, the fine gradings were classified in [Eld10]. For the exceptional simple algebras they were classified in [DM06] and [BT09] for G 2 , in [DM09] for F 4 and in [DV] for E 6 .…”

Section: Introductionmentioning

confidence: 99%

Fine gradings and gradings by root systems on simple Lie algebras

Elduque¹

2015

Rev. Mat. Iberoam.

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Abstract. Given a fine abelian group grading Γ : L = g∈G Lg of a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero, with G being the universal grading group, it is shown that the induced grading by the free group G/ tor(G) on L is a grading by a (not necessarily reduced) root system. Some consequences for the classification of fine gradings on the exceptional simple Lie algebras are drawn.

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