Extensions of <i>n</i>-Hom Lie algebras

doi:10.1007/s11464-014-0372-8

Front. Math. China

2015, Vol. 10

Issue (3) : 511-522 https://doi.org/10.1007/s11464-014-0372-8

RESEARCH ARTICLE

Extensions of n-Hom Lie algebras

Ruipu BAI(

),Ying LI

College of Mathematics and Computer Science, Hebei University, Baoding 071002, China

Download: PDF(124 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

n-Hom Lie algebras are twisted by n-Lie algebras by means of twisting maps. n-Hom Lie algebras have close relationships with statistical mechanics and mathematical physics. The paper main concerns structures and representations of n-Hom Lie algebras. The concept of n_ρ-cocycle for an n-Hom Lie algebra (G, [,… , ], α) related to a G-module (V, ρ, β) is proposed, and a sufficient condition for the existence of the dual representation of an n-Hom Lie algebra is provided. From a G-module (V, ρ, β) and an n_ρ-cocycle θ, an n-Hom Lie algebra (T_θ(V ), [, … , ]θ, γ) is constructed on the vector space T_θ(V ) = G⊕V, which is called the T_θ-extension of an n-Hom Lie algebra (G, [, … , ], α) by the G-module (V, ρ, β).

Keywords n-Hom Lie algebra representation extension n_ρ-cocycle

Corresponding Author(s): Ruipu BAI

Issue Date: 01 April 2015

Cite this article:

Ruipu BAI,Ying LI. Extensions of n-Hom Lie algebras[J]. Front. Math. China, 2015, 10(3): 511-522.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-014-0372-8
https://academic.hep.com.cn/fmc/EN/Y2015/V10/I3/511

1	Ammar F, Mabrouk S, Makhlouf A. Representations and cohomology of n-ary multiplicative Hom-Nambu-Lie algebras. J Geom Phys, 2011, 61(10): 1898-1913 https://doi.org/10.1016/j.geomphys.2011.04.022
2	Ataguema H, Makhlouf A, Silvestrov S. Generalization of n-ary Nambu algebras and beyond. J Math Phys, 2009, 50(8): 083501 https://doi.org/10.1063/1.3167801
3	Azcarraga J, Izquierdo J. n-ary algebras: a review with applications. J Phys A, 2010, 43(29): 293001 https://doi.org/10.1088/1751-8113/43/29/293001
4	Bagger J, Lambert N. Gauge symmetry and supersymmetry of multiple M2-branes. Phys Rev, 2008, D77(6): 065008 https://doi.org/10.1103/PhysRevD.77.065008
5	Bai R, Bai C, Wang J. Realizations of 3-Lie algebras. J Math Phys, 2010, 51(6): 063505 https://doi.org/10.1063/1.3436555
6	Bai R, Li Y. Tθ*-Extensions of n-Lie Algebras. ISRN Algebra, 2011, V2011 https://doi.org/10.5402/2011/381875
7	Bai R, Li Y, Wu W. Extensions of n-Lie algebras. Sci Sin Math, 2012, 42(6): 1-10 (in Chinese)
8	Bai R, Wang J, Li Z. Derivations of the 3-Lie algebra realized by gl(n,C). J Nonlinear Math Phys, 2011, 18(1): 151-160
9	Baxter R. Partition function for the eight-vertex lattice model. Ann Physics, 1972, 70: 193-228 https://doi.org/10.1016/0003-4916(72)90335-1
10	Baxter R. Exactly Solved Models in Statistical Mechanics. London: Academic Press, 1982
11	Benayadi S, Makhlouf A. Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. arXiv: 1009.4225
12	Filippov V. n-Lie algebras. Sibirsk Mat Zh, 1985, 26(6): 126-140
13	Hartwig J, Larsson D, Silvestrov S. Deformations of Lie algebras using σ-derivations. J Algebra, 2006, 295(2): 314-361 https://doi.org/10.1016/j.jalgebra.2005.07.036
14	Kasymov S. On a theory of n-Lie algebras. Algebra Logika, 1987, 26(3): 277-297 https://doi.org/10.1007/BF02009328
15	Marmo G, Vilasi G, Vinogradov A. The local structure of n-Poisson and n-Jacobi manifolds. J Geom Phys, 1998, 25(1-2): 141-182 https://doi.org/10.1016/S0393-0440(97)00057-0
16	Nambu Y. Generalized Hamiltonian dynamics. Phys Rev, 1973, D7: 2405-2412 https://doi.org/10.1103/PhysRevD.7.2405
17	Okubo S. Introduction to Octonion and Other Non-Associative Algebras in Physics. Cambridge: Cambridge Univ Press, 1995 https://doi.org/10.1017/CBO9780511524479
18	Pozhidaev A. Representations of vector product n-Lie algebras. Comm Algebra, 2004, 32(9): 3315-3326 https://doi.org/10.1081/AGB-120038644
19	Yang C. Some exact results for the many-body problem in one dimension with replusive delta-function interaction. Phys Rev Lett, 1967, 19(23): 1312-1315 https://doi.org/10.1103/PhysRevLett.19.1312
20	Yau D. On n-ary hom Nambu and Hom-Nambu-Lie algebras. J Geom Phys, 2012, 62(2): 506-522 https://doi.org/10.1016/j.geomphys.2011.11.006
21	Yau D. A Hom-associative analogue of n-ary Hom-Nambu algebras. arXiv: 1005.2373

[1]	Lifang WANG, Ke WU, Jie YANG, Zifeng YANG. Representation of elliptic Ding-Iohara algebra[J]. Front. Math. China, 2020, 15(1): 155-166.
[2]	Jiashun HU, Xiang MA, Chunxiong ZHENG. Global geometrical optics method for vector-valued Schrödinger problems[J]. Front. Math. China, 2018, 13(3): 579-606.
[3]	Peng YU. A recollement construction of Gorenstein derived categories[J]. Front. Math. China, 2018, 13(3): 691-713.
[4]	Lei LIU. Lagrangian Grassmann manifold Λ(2)[J]. Front. Math. China, 2018, 13(2): 341-365.
[5]	Zhihua WANG, Libin LI. Double Frobenius algebras[J]. Front. Math. China, 2018, 13(2): 399-415.
[6]	Zhi CHENG. Trivial extension of Koszul algebras[J]. Front. Math. China, 2017, 12(5): 1045-1056.
[7]	Zhishan YANG. Ideal counting function in cubic fields[J]. Front. Math. China, 2017, 12(4): 981-992.
[8]	Huanhuan LI, Yunge XU. Koszulity and Koszul modules of dual extension algebras[J]. Front. Math. China, 2017, 12(3): 583-596.
[9]	Hiroaki KOMATSU. On the generalized derivations of bimodules[J]. Front. Math. China, 2017, 12(1): 135-142.
[10]	Jin CHENG. *Generalized B(m, n), C(n),D(m, n)-graded Lie superalgebras arising from fermionic-bosonic representations*[J]. Front. Math. China, 2016, 11(6): 1451-1470.
[11]	Daiki OBARA. Hochschild cohomology ring modulo nilpotence of a one point extension of a quiver algebra defined by two cycles and a quantum-like relation[J]. Front. Math. China, 2016, 11(4): 1003-1015.
[12]	Haiyan ZHU. Constructing cotorsion pairs over generalized path algebras[J]. Front. Math. China, 2016, 11(4): 1079-1096.
[13]	Claus Michael RINGEL. Representation theory of Dynkin quivers. Three contributions[J]. Front. Math. China, 2016, 11(4): 765-814.
[14]	Chan Yong HONG,Nam Kyun KIM,Tai Keun KWAK,Yang LEE. On extensions of matrix rings with skew Hochschild 2-cocycles[J]. Front. Math. China, 2016, 11(4): 869-900.
[15]	Yongjie WANG,Shikui SHANG,Yun GAO. Color cyclic homology and Steinberg Lie color algebras[J]. Front. Math. China, 2015, 10(5): 1179-1202.

Viewed

Full text

Abstract

Cited

Shared

Discussed