Constructions of metric (n + 1)-Lie algebras

Abstract

Metric n-Lie algebras have wide applications in mathematics and mathematical physics. In this paper, the authors introduce two methods to construct metric (n+1)-Lie algebras from metric n-Lie algebras for n ≥ 2. For a given m-dimensional metric n-Lie algebra (g, [, · · ·, ], B _g), via one and two dimensional extensions L = g +Fc and g₀ = g + Fx ⁻¹ +Fx ⁰ of the vector space g and a certain linear function f on g, we construct (m+1)- and (m+2)-dimensional (n+1)-Lie algebras (L, [, · · ·, ] _cf) and (g₀, [, · · ·, ]1), respectively. Furthermore, if the center Z(g) is non-isotropic, then we obtain metric (n+1)-Lie algebras (L, [, · · ·, ] _cf, B) and (g₀, [, · · ·, ]₁, B) which satisfy B|_g×g = B _g. Following this approach the extensions of all (n + 2)-dimensional metric n-Lie algebras are discussed.

Keywords

n-Lie algebra / Metric n-Lie algebra / Extension / Isotropic center

References

[1] Filippov V. T.. n-Lie algebras. Sib. Mat. Zh., 1985, 26: 126-140

[2] Nambu Y.. Generalized Hamiltonian dynamics. Phys. Rev. D, 1973, 7: 2405-2412

[3] Takhtajan L.. On foundation of the generalized Nambu mechanics. Comm. Math. Phys., 1994, 160: 295-315

[4] Bagger J., Lambert N.. Gauge symmetry and supersymmetry of multiple M2-branes. Phys. Rev. D, 2008, 77: 065008

[5] Ho, P., Hou, R. and Matsuo, Y., Lie 3-algebra and multiple M2-branes, arXiv: 0804.2110.

[6] Ho P., Chebotar M., Ke W.. On skew-symmetric maps on Lie algebras. Proc. Royal Soc. |Edinburgh^A, 2003, 113: 1273-1281

[7] Gustavsson, A., Algebraic structures on parallel M2-branes, arXiv: 0709.1260.

[8] Papadopoulos, G., M2-branes, 3-Lie algebras and Plucker relations, arXiv: 0804.2662.

[9] Hoppe J.. On M-algebras, the quantisation of Nambu-mechanics, and volume preserving diffeomorphisms. Helv. Phys. Acta, 1997, 70: 302-317

[10] Ling, W., On the structure of n-Lie algebras, University-GHS-Siegen, 1993.

[11] Kasymov S.. On a theory of n-Lie algebras. Algebra i Logika, 1987, 26: 277-297

[12] de Azcrraga, J. A. and Izquierdo, J. M., n-ary algebras: A review with applications, J. Phys. A: Math. Theor., 43, 2010, 293001, arXiv: 1005.1028[math-ph].

[13] de Azcrraga J. A., Izquierdo J. M.. Cohomology of Filippov algebras and an analogue of Whitehead’s lemma. J. Phys. Conf. Ser., 2009, 175: 1-24

[14] Williams M. P.. Nilpotent N-Lie algebras, 2004

[15] Bai R., Shen C., Zhang Y.. 3-Lie algebras with an ideal N. Linear Alg. Appl., 2009, 431: 673-700

[16] Bai R., Han W., Bai C.. The generating index of an n-Lie algebra. J. Phys. A: Math. Theor., 2011, 44: 185201

[17] Dzhumadildaev, A. S., Identities and derivations for Jacobian algebras, arXiv: 0202040v1[math. RA].

[18] Medeiros P., Figueroa-O’Farrill J., Mndez-Escobar E., Ritter P.. Metric 3-Lie algebras for unitary Bagger-Lambert theories, J. High Energy Phys., 2009

[19] Jin Y., Liu W., Zhang Z.. Metric n-Lie algebras. Commn. Algebra, 2011, 39(2): 572-583

[20] Jin, Y., Liu, W. and Zhang, Z., Real simple n-Lie algebras admitting metric structures, J. Phys. A: Math. Theor., 42(48), 2009.

[21] Bai R., Wu W., Li Z.. Some results on metric n-Lie algebras, Acta Methematics Sinica. English Series, 2012, 28(6): 1209-1220

[22] Pozhidaev A. P.. Monomial n-Lie algebras. Algebra and Logic, 1998, 37(5): 307-322

[23] Pozhidaev A. P.. Simple n-Lie algebras. Algebra and Logic, 1999, 38(3): 181-192

[24] Ho P., Imamura Y., Matsuo Y.. M2 to D2 revisited, J. High Energy Phys., 2008

[25] Bai R., Bai C., Wang J.. Realizations of 3-Lie algebras. J. Math. Phys., 2010, 51: 063505

[26] Bai, R. and Wu, Y., Constructing 3-Lie algebras, arXiv:1306.1994v1[math-ph].

[27] Bai R., Wu W., Li J., Zhou H.. Constructing (n + 1)-Lie algebras from n-Lie algebras. J. Phys. A: Math. Theor., 2012, 45: 475206

[28] Bai R., Han W., Liu H.. Structures of 2-step nilpotent 3-Lie algebras. Antarctica J. Math., 2012, 9(1): 23-40

[29] Bai R., Song G., Zhang Y.. On classification of n-Lie algebras. Front. Math. China, 2011, 6(4): 581-606