Leibniz Algebras Admitting a Multiplicative Basis
Tóm tắt
In the literature, many of the descriptions of different classes of Leibniz algebras
$$(L, [\cdot , \cdot ])$$
have been made by giving the multiplication table on the elements of a basis
$${{\mathcal {B}}}=\{v_{k}\}_{k \in K}$$
of L, in such a way that for any
$$i,j \in K$$
we have that
$$[v_i,v_j] = \lambda _{i,j} [v_j,v_i]\in {{\mathbb {F}}} v_k$$
for some
$$k \in K$$
, where
$${{\mathbb {F}}}$$
denotes the base field and
$$\lambda _{i,j} \in {{\mathbb {F}}}$$
. In order to give an unifying viewpoint of all these classes of algebras, we introduce the more general category of Leibniz algebras admitting a multiplicative basis and study its structure. We show that if a Leibniz algebra L admits a multiplicative basis, then it is the direct sum
$$L=\bigoplus \nolimits _{\alpha }{{{\mathcal {I}}}}_{\alpha }$$
with any
$${{\mathcal {I}}}_{\alpha }$$
a well-described ideal of L admitting a multiplicative basis inherited from
$${\mathcal {B}}$$
. Also the
$${{\mathcal {B}}}$$
-simplicity of L is characterized in terms of the multiplicative basis.
Tài liệu tham khảo
Abdykassymova, S., Dzhumaldil’daev, A.: Leibniz algebras in characteristic \(p\). Comptes Rendus Acad. Sci. Paris Ser. I Math. 332(12), 1047–1052 (2001)
Albeverio, S., Ayupov, Sh.A., Omirov, B.A.: On nilpotent and simple Leibniz algebras. Commun. Algebra 33(1), 159–172 (2005)
Albeverio, S., Ayupov, ShA, Omirov, B.A., Khudoyberdiyev, AKh: \(n\)-Dimensional filiform Leibniz algebras of length \((n-1)\) and their derivations. J. Algebra 319(6), 2471–2488 (2008)
Albeverio, S., Omirov, B.A., Rakhimov, I.S.: Varieties of nilpotent complex Leibniz algebras of dimension less than five. Commun. Algebra 33(5), 1575–1585 (2005)
Ayupov, ShA, Omirov, B.A.: On some classes of nilpotent Leibniz algebras. Sib. Math. J. 42(1), 18–29 (2001)
Bloh, A.: On a generalization of the concept of Lie algebra. Dokl. Akad. Nauk SSSR 165, 471–473 (1965)
Bloh, A.: Cartan–Eilenberg homology theory for a generalized class of Lie algebras. Dokl. Akad. Nauk SSSR 175, 266–268 (1967). Translated as Sov. Math. Dokl. 8, 824–826 (1967)
Bloh, A.: A certain generalization of the concept of Lie algebra. Algebra and number theory. Moskov. Gos. Ped. Inst. Ucen. Zap. No. 375, 9–20 (1971)
Cabezas, J.M., Camacho, L.M., Rodriguez, I.M.: On filiform and 2-filiform Leibniz algebras of maximum length. J. Lie Theory 18(2), 335–350 (2008)
Calderón, A.J.: On the structure of graded Lie algebras. J. Math. Phys. 50(10), 103513, 8 (2009)
Calderón, A.J., Sánchez, J.M.: On the structure of graded Lie superalgebras. Mod. Phys. Lett. A 27(25), 1250142 (2012)
Calderón, A.J., Sánchez, J.M.: Split Leibniz algebras. Linear Algebra Appl. 436(6), 1648–1660 (2012)
Calderón, A.J., Sánchez, J.M.: On the structure of split Leibniz superalgebras. Linear Algebra Appl. 438, 4709–40725 (2013)
Camacho, L.M., Casas, J.M., Gómez, J.R., Ladra, M., Omirov, B.A.: On nilpotent Leibniz n-algebras. J. Algebra Appl. 11(3), 1250062, 17 (2012)
Camacho, L.M., Cañete, E.M., Gómez, J.R., Omirov, B.A.: 3-filiform Leibniz algebras of maximum length, whose naturally graded algebras are Lie algebras. Linear Multilinear Algebra 59(9), 1039–1058 (2011)
Camacho, L.M., Cañete, E.M., Gómez, J.R., Redjepov, ShB: Leibniz algebras of nilindex \(n-3\) with characteristic sequence \((n-3,2,1)\). Linear Algebra Appl. 438(4), 1832–1851 (2013)
Camacho, L.M., Gomez, J.R., González, A.J., Omirov, B.A.: Naturally graded 2-filiform Leibniz algebras. Commun. Algebra 38, 3671–3685 (2010)
Camacho, L.M., Gómez, J.R., González, A.J., Omirov, B.A.: The classification of naturally graded p-filiform Leibniz algebras. Commun. Algebra 39(1), 153–168 (2011)
Camacho, L.M., Gómez, J.R., Omirov, B.A.: Naturally graded \((n-3)\)-filiform Leibniz algebras. Linear Algebra Appl. 433(2), 433–446 (2010)
Casas, J.M., Khudoyberdiyev, AKh, Ladra, M., Omirov, B.A.: On the degenerations of solvable Leibniz algebras. Linear Algebra Appl. 439(2), 472–487 (2013)
Casas, J.M., Ladra, M., Omirov, B.A., Karimjanov, I.A.: Classification of solvable Leibniz algebras with null-filiform nilradical. Linear Multilinear Algebra 61(6), 758–774 (2013)
Casas, J.M., Ladra, M., Omirov, B.A., Karimjanov, I.A.: Classification of solvable Leibniz algebras with naturally graded filiform nilradical. Linear Algebra Appl. 438(7), 2973–3000 (2013)
Cañete, E.M., Khudoyberdiyev, AKh: The classification of 4-dimensional Leibniz algebras. Linear Algebra Appl. 439(1), 273–288 (2013)
Fialowski, A., Khudoyberdiyev, AKh, Omirov, B.A.: A characterization of nilpotent Leibniz algebras. Algebras Represent. Theory 16(5), 1489–1505 (2013)
Ladra, M., Omirov, B.A., Rozikov, U.A.: Classification of p-adic 6-dimensional filiform Leibniz algebras by solutions of \(x^q=a\). Cent. Eur. J. Math. 11(6), 1083–1093 (2013)
Liu, D., Hu, N.: Leibniz algebras graded by finite root systems. Algebra Colloq. 17(3), 431–446 (2010)
Loday, J.L.: Une version non commutative des algébres de Lie: les algébres de Leibniz. L’Ens. Math. 39, 269–293 (1993)
Omirov, B.A., Rakhimov, I.S., Turdibaev, R.M.: On description of Leibniz algebras corresponding to \({sl}_2\). Algebras Represent. Theory 16, 1507–1519 (2013)
Rakhimov, I.S., Al-Nashri, A.-H.: On derivations of some classes of Leibniz algebras. J. Gen. Lie Theory Appl. 6, Article ID G120501 (2012)
Rakhimov, I.S., Atan Kamel, A.M.: On contractions and invariants of Leibniz algebras. Bull. Malays. Math. Sci. Soc. (2) 35(2A), 557–565 (2012)
Rakhimov, I.S., Hassan, M.A.: On low-dimensional filiform Leibniz algebras and their invariants. Bull. Malays. Math. Sci. Soc. (2) 34(3), 475–485 (2011)
Rakhimov, I.S., Sozan, J.: Description of nine dimensional complex filiform Leibniz algebras arising from naturally graded non Lie filiform Leibniz algebras. Int. J. Algebra 5, 271–280 (2009)