On Naturally Graded Lie and Leibniz Superalgebras
- PDF / 464,166 Bytes
- 25 Pages / 439.37 x 666.142 pts Page_size
- 51 Downloads / 226 Views
On Naturally Graded Lie and Leibniz Superalgebras L. M. Camacho1
· R. M. Navarro2 · J. M. Sánchez3
Received: 8 April 2019 / Revised: 5 November 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019
Abstract In general, the study of gradations has always represented a cornerstone in the study of non-associative algebras. In particular, natural gradation can be considered to be the first and most relevant gradation of nilpotent Leibniz (resp. Lie) algebras. In fact, many families of relevant solvable Leibniz (resp. Lie) algebras have been obtained by extensions of naturally graded algebras, i.e., solvable algebras with a well-structured nilradical. Thus, the aim of this work is introducing the concept of natural gradation for Lie and Leibniz superalgebras. Moreover, after having defined naturally graded Lie and Leibniz superalgebras, we characterize natural gradations on a very important class of each of them, that is, those with maximal supernilindex. Keywords Lie (super)algebras · Cohomology · Deformation · Leibniz (super)algebras · Naturally graded Mathematics Subject Classification 17A32 · 17B30 · 17B70 · 17A70
Communicated by Peyman Niroomand. This work has been supported by Agencia Estatal de Investigación (Spain), Grant MTM2016-79661-P (European FEDER support included, UE), by the PCI of the UCA “Teoría de Lie y Teoría de Espacios de Banach” and by the PAI with project number FQM298.
B
L. M. Camacho [email protected] R. M. Navarro [email protected] J. M. Sánchez [email protected]
1
Dpto. Matemática Aplicada I, Universidad de Sevilla, Sevilla, Spain
2
Dpto. de Matemáticas, Universidad de Extremadura, Cáceres, Spain
3
Dpto. de Matemáticas, Universidad de Cádiz, Campus de Puerto Real, Cádiz, Spain
123
L. M. Camacho et al.
1 Introduction In general, the study of graded algebras has always played a fundamental role into Lie theory (see for instance [10,15]). Recently, the importance of naturally graded Lie and Leibniz algebras has been increased by means of the use of them as nilradical of relevant families of solvable ones (see [1,5,13,28]). Therefore, it remains as an immediate and future work and the use of the results of the present paper to obtain important families of solvable Lie and Leibniz superalgebras by extensions of nonnilpotent outer derivations. Recall that filiform Lie algebras were firstly introduced in [34] and the generalization for Lie superalgebras has already been obtained (see [8]) and, in the same way as occurs for Lie algebras, filiform Lie superalgebras have maximal supernilindex. On the other hand, the notion of Leibniz superalgebras as a generalization of Leibniz algebras was firstly introduced in [2], and general graded Leibniz algebras were considered before in work [29], though. Since Leibniz algebras are a generalization of Lie algebras [30], consequently many of the features of Leibniz superalgebras are generalization of Lie superalgebras [3,11,12,19]. Likewise, the study of nilpotent Leibniz algebras [2,6,7] can be very useful to
Data Loading...
