Corrigendum to “Filiform color Lie superalgebras” [J. Geom. Phys. 61 (2011) 8–17]

Journal of Geometry and Physics 62 (2012) 1572–1573

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Erratum

Corrigendum to ‘‘Filiform color Lie superalgebras’’ [J. Geom. Phys. 61 (2011) 8–17] Yu. Khakimdjanov a , R.M. Navarro b,∗ a

Laboratoire de mathématiques et applications, Université de Haute Alsace, Mulhouse, France

b

Dpto. de Matemáticas, Universidad de Extremadura, Cáceres, Spain

article

info

Article history: Received 25 January 2012 Accepted 30 January 2012 Available online 7 February 2012

Remark 5.7 and Example (1) on page 14 must be rewritten, as follows: Remark 5.7. In the particular case of G = Z4 the theorem for the adapted basis rests on the following for L = L0 ⊕ Ł1 ⊕L2 ⊕ L3 ∈ F n,m,p,r :

 [X0 , Xi ] = Xi+1 ,      [X0 , Xn−1 ] = 0,      [X0 , Yj ] = Yj+1 ,    [X0 , Ym ] = 0, [X0 , Zk ] = Zk+1 ,      [X0 , Zp ] = 0,     [X , W ] = Wl+1 ,    0 l [X0 , Wr ] = 0

1 ≤ i ≤ n − 2, 1 ≤ j ≤ m − 1, 1 ≤ k ≤ p − 1, 1 ≤ l ≤ r − 1,

with {X0 , X1 , . . . , Xn−1 } a basis of L0 , {Y1 , . . . , Ym } a basis of L1 and {Z1 , . . . , Zp }, {W1 , . . . , Wr } bases of L2 and L3 respectively. It is not difficult to see that for G = Z4 , there is only one non-degenerate possibility for β , i.e. if β is a commutation factor β : Z4 × Z4 −→ F \ {0}, then β(g , h) = (−1)gh or β(g , h) = 1 (degenerate case) ∀g , h ∈ Z4 = {0, 1, 2, 3}. Thus, the anticommutative identity and the Jacobi identity for L = L0 ⊕ L1 ⊕ L2 ⊕ L3 rest on: (1) [X , Y ] = −(−1)gh [Y , X ]∀X ∈ Lg , Y ∈ Lh ; (2) [[X , Y ], Z ] = [X , [Y , Z ]] − (−1)gh [Y , [X , Z ]]∀X ∈ Lg , Y ∈ Lh , Z ∈ L. Examples. Other examples of filiform color Lie superalgebras distinct from the model are easy to obtain. Thus: (1) If G = Z4 we have a whole family of non-model filiform color Lie superalgebras with β(g , h) = (−1)gh as the commutation factor: µn,m,p,r with n, m, r ≥ 2 and p ≥ 3, that can be expressed using an adapted basis {X0 , X1 , . . . ,

∗

DOI of original article: 10.1016/j.geomphys.2010.09.004. Corresponding author. Tel.: +34 927257213; fax: +34 927257203. E-mail addresses: [email protected] (Yu. Khakimdjanov), [email protected] (R.M. Navarro).

0393-0440/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2012.01.018

Yu. Khakimdjanov, R.M. Navarro / Journal of Geometry and Physics 62 (2012) 1572–1573

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Xn−1 , Y1 , . . . , Ym , Z1 , . . . , Zp , W1 , . . . , Wr } via the following non-null bracket products:

µn,m,p,r

 [X0 , Xi ] = Xi+1 ,     [ X0 , Yj ] = Yj+1 ,      [  X0 , Zk ] = Zk+1 = [X0 , Wl ] = Wl+1    [Y1 , Y1 ] = Zp       [Z1 , Z2 ] = Xn−1 [Y1 , Z1 ] = Wr

1≤i≤n−2 1≤j≤m−1 1≤k≤p−1 1≤l≤r −1

and we observe that the product [Y1 , Y1 ] = Zp is symmetric and, in particular, for n = 2, L0 is the abelian Lie algebra.