Corrigendum to “Integrable deformations of nilpotent color Lie superalgebras” [J. Geom. Phys. 61 (2011) 1797–1808]

Journal of Geometry and Physics 62 (2012) 1571

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Erratum

Corrigendum to ‘‘Integrable deformations of nilpotent color Lie superalgebras’’ [J. Geom. Phys. 61 (2011) 1797–1808] 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 26 January 2012 Accepted 30 January 2012 Available online 14 March 2012

In the paper, (Z3 , β)-color Lie superalgebras are considered. All the errors in the paper arise from considering β(g , h) = (−1)gh , g , h ∈ Z3 = {0, 1, 2}, as a commutation factor; in fact, the unique admissible commutation factor is exactly β(g , h) = 1, ∀g , h. Thus, (−1)gh must be replaced by 1 throughout the paper. If we consider the model filiform (Z3 , β)color Lie superalgebra Ln,m,p , then all the bracket products are anti-commutative ([X , Y ] = −[Y , X ]). As a consequence, all the 2-cocycles Z02 (Ln,m,p ; Ln,m,p ) are, in particular, skew-symmetric bilinear mappings; thus the subspace of 2-cocycles D is exactly Z 2 (L; L) ∩ Hom(L1 ∧ L1 , L2 ). Of the three theorems of the paper, the one that is affected is Theorem 1, that must be rewritten as follows: Theorem 1. If D = Z 2 (L; L) ∩ Hom(L1 ∧ L1 , L2 ), then we have the following values for the dimension of D:

dim D =

 m(m − 1)    2   1   (4mp − p2 − 2p − 1)    8 1   (4mp − p2 − 2p + 3)   8       1 (4mp − p2 − 2p) 8

∗

if p ≥ 2m − 1 if p < 2m − 1,

p ≡ 1(mod 4) and m odd, or p ≡ 3(mod 4) and m even

if p < 2m − 1,

p ≡ 3(mod 4) and m odd, or p ≡ 1(mod 4) and m even

if p < 2m − 1

and p even.

DOI of original article: 10.1016/j.geomphys.2011.03.019. 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.2012.01.017