Fine group gradings of the real forms of $sl(4,\C)$, $sp(4,\C)$, and $o(4,\C)$

Edita Pelantov\'a; Ji\v{r}\'i Patera; Milena Svobodov\'a

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Fine group gradings of the real forms of $sl(4,\C)$, $sp(4,\C)$, and $o(4,\C)$

Jiří Patera, Edita Pelantová, Milena Svobodová

Mathematical Physics · Submitted 14 Aug 2006

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The authors' main claim 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 $o(4,\C)$.

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arXiv IDmath-ph/0608032 (version 2)

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CitationEdita Pelantov\'a, Ji\v{r}\'i Patera, Milena Svobodov\'a. "Fine group gradings of the real forms of $sl(4,\C)$, $sp(4,\C)$, and $o(4,\C)$." arXiv:math-ph/0608032 (2006). https://pith.science/paper/MZTSKXIM

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.06 · mentions code or data availability 0.02.

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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 $sl(4,\C)\supset sp(4,\C)\supset o(4,\C)$ are an important tool in our presentation. Systematic use is made of the faithful representations of the three Lie algebras by $4\times 4$ matrices.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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 $o(4,\C)$.	contribution 0.97	0.95	0.78	asserted 1.00
s1	All together 12 real Lie algebras are considered, and the total of 44 of their fine group gradings are listed.	result 0.90	0.93	0.29	asserted 1.00
s2	The inclusions $sl(4,\C)\supset sp(4,\C)\supset o(4,\C)$ are an important tool in our presentation.	method 0.99	0.57	0.04	asserted 1.00
s3	Systematic use is made of the faithful representations of the three Lie algebras by $4\times 4$ matrices.	method 1.00	0.88	0.02	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from proof or derivation 1.00.

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Original 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 $sl(4,\C)\supset sp(4,\C)\supset o(4,\C)$ are an important tool in our presentation. Systematic use is made of the faithful representations of the three Lie algebras by $4\times 4$ matrices.

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2 arXiv papers cite this in 8 sentences

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Contests it 1 paper

It is only by mistake that the Cartan fine group grading of $usp(2,2$) is missing in [ref] – the respective MAD-group was forgotten there; hence the result in [ref] is richer by this one fine group grading.

Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms · $B_2=o(5,ℂ)$, $C_2=sp(4,ℂ)$ · contest 0.93

Cites it as background 1 paper

As it was already pointed out by Patera et al. in a series of papers on the subject [ref] , gradings are in the background of the choice of bases and the additive quantum numbers, and are a key ingredient when studying contractions that keep “undeformed" certain subalgebra (e.g. see [ref] ).

Gradings on $o(8,\mathbb C)$ · Introduction · 0.96

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