Geodesic
A generalization of Lie $H$-pseudo-bialgebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 1, pp. 3-22 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We investigate Hom–Lie $H$-pseudo-bialgebras. We present some examples and a theorem that allows constructing these new algebraic structures. We consider coboundary Hom–Lie $H$-pseudo-bialgebras and the corresponding classical Hom–Yang–Baxter equations.
Keywords: Hom–Lie bialgebra, classical Hom–Yang–Baxter equation, Hom-algebra.
Mots-clés : Hom–Lie $H$-pseudo-bialgebra 
@article{TMF_2017_192_1_a0,
     author = {Qinxiu Sun and Fang Li},
     title = {A~generalization of {Lie} $H$-pseudo-bialgebras},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {3--22},
     year = {2017},
     volume = {192},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2017_192_1_a0/}
}
TY  - JOUR
AU  - Qinxiu Sun
AU  - Fang Li
TI  - A generalization of Lie $H$-pseudo-bialgebras
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2017
SP  - 3
EP  - 22
VL  - 192
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2017_192_1_a0/
LA  - ru
ID  - TMF_2017_192_1_a0
ER  - 
%0 Journal Article
%A Qinxiu Sun
%A Fang Li
%T A generalization of Lie $H$-pseudo-bialgebras
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2017
%P 3-22
%V 192
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2017_192_1_a0/
%G ru
%F TMF_2017_192_1_a0
Qinxiu Sun; Fang Li. A generalization of Lie $H$-pseudo-bialgebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 1, pp. 3-22. http://geodesic.mathdoc.fr/item/TMF_2017_192_1_a0/

[1] B. Bakalov, A. D'Andrea, V. G. Kac, “Theory of finite pseudoalgebras”, Adv. Math., 162:1 (2001), 1–140 | DOI | MR | Zbl

[2] A. D'Andrea, V. G. Kac, “Structure theory of finite conformal algebras”, Selecta Math. (N. S.), 4:3 (1998), 377–418 | DOI | MR | Zbl

[3] V. G. Kats, Verteksnye algebry dlya nachinayuschikh, MTsNMO, M., 2005

[4] E. Cartan, “Les groupes de transformation continus, infinis, simples”, Ann. Sci. École Norm. Sup. (3), 26 (1909), 93–161 | MR | Zbl

[5] V. Guillemin, “A Jordan–Hölder decomposition for a certain class of infinite dimensional Lie algebras”, J. Differential Geom., 2:3 (1968), 313–345 | DOI | MR | Zbl

[6] V. Guillemin, “Infinite-dimensional primitive Lie algebras”, J. Differential Geom., 4:3 (1970), 257–282 | DOI | MR | Zbl

[7] I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Nonlinear Science: Theory and Applications, John Wiley and Sons, Chichester, 1993 | MR

[8] B. A. Dubrovin, S. P. Novikov, “Gidrodinamika slabo deformirovannykh solitonnykh reshetok. Differentsialnaya geometriya i gamiltonova teoriya”, UMN, 44:6(270) (1989), 29–98 | DOI | MR | Zbl

[9] I. M. Gelfand, I. Ya. Dorfman, “Gamiltonovy operatory i klassicheskoe uravnenie Yanga–Bakstera”, Funkts. analiz i ego pril., 16:4 (1982), 1–9 | DOI | MR | Zbl

[10] O. I. Mokhov, “O skobkakh Puassona tipa Dubrovina–Novikova (DN-skobki)”, Funkts. analiz i ego pril., 22:4 (1988), 92–93 | DOI | MR | Zbl

[11] X. Xu, “Equivalence of conformal superalgebras to Hamiltonian superoperators”, Algebra Colloq., 8:1 (2001), 63–92 | MR | Zbl

[12] E. I. Zelmanov, “Ob odnom klasse lokalnykh translyatsionno invariantnykh algebr Li”, Dokl. AN SSSR, 292:6 (1987), 1294–1297 | MR | Zbl

[13] N. Aizawa, H. Sato, “q-deformation of the Virasoro algebra with central extension”, Phys. Lett. B, 256:2 (1991), 185–190 | DOI | MR | Zbl

[14] C. Daskaloyannis, “Generalized deformed Virasoro algebras”, Modern Phys. Lett. A, 7:9 (1992), 809–816 | DOI | MR | Zbl

[15] N. H. Hu, “q-Witt algebras, q-Lie algebras, q-holomorph structure and representations”, Algebra Colloq., 6:1 (1999), 51–70 | MR | Zbl

[16] H. Ataguema, A. Makhlouf, S. Silvestrov, “Generalization of $n$-ary Nambu algebras and beyond”, J. Math. Phys., 50:8 (2009), 083501, 15 pp. | DOI | MR | Zbl

[17] J. Arnlind, A. Makhlouf, S. Silvestrov, “Ternary Hom–Nambu–Lie algebras induced by Hom–Lie algebras”, J. Math. Phys., 51:4 (2010), 043515, 11 pp. | DOI | MR | Zbl

[18] J. T. Hartwig, D. Larsson, S. D. Silvestrov, “Deformations of Lie algebras using $\sigma$-derivations”, J. Algebra, 295:2 (2006), 314–361 | DOI | MR | Zbl

[19] A. Makhlouf, S. D. Silvestrov, “Hom–algebra structures”, J. Gen. Lie Theory Appl., 2:2 (2008), 51–64 | DOI | MR | Zbl

[20] A. Makhlouf, S. Silvestrov, “Hom–algebras and Hom–coalgebras”, J. Algebra Appl., 9:4 (2010), 553–589, arXiv: 0811.0400 | DOI | MR

[21] A. Makhlouf, S. D. Silvestrov, “Hom–Lie admissible Hom–coalgebras and Hom–Hopf algebras”, Generalized Lie Theory in Mathematics, Physics and Beyond, eds. S. Silvestrov, E. Paal, V. Abramov, A. Stolin, Springer, Berlin, 2009, 189–206 | DOI | MR | Zbl

[22] D. Yau, “On $n$-ary Hom–Nambu and Hom–Nambu–Lie algebras”, J. Geom. Phys., 62:2 (2012), 506–522, arXiv: 1004.2080 | DOI | MR

[23] F. Ammar, Z. Ejbehi, A. Makhlouf, “Cohomology and deformations of Hom–algebras”, J. Lie Theory, 21:4 (2011), 813–836, arXiv: 1005.0456 | MR

[24] F. Ammar, S. Mabrouk, A. Makhlouf, “Representations and cohomology of $n$-ary multiplicative Hom–Nambu–Lie algebras”, J. Geom. Phys., 61:10 (2011), 1898–1913 | DOI | MR | Zbl

[25] A. Makhlouf, S. Silvestrov, “Notes on 1-parameter formal deformations of Hom–associative and Hom–Lie algebras”, Forum Math., 22:4 (2010), 715–739, arXiv: 0712.3130 | DOI | MR

[26] D. Yau, “Hom–algebras and homology”, J. Lie Theory, 19:2 (2009), 409–421 | MR | Zbl

[27] D. Yau, “The Hom–Yang–Baxter equation, Hom–Lie algebras, and quasi-triangular bialgebras”, J. Phys. A: Math. Theor., 42:16 (2009), 165202, 12 pp. | DOI | MR | Zbl

[28] D. Yau, “The Hom–Yang–Baxter equation and Hom–Lie algebras”, J. Math. Phys., 52:5 (2011), 053502, 19 pp. | DOI | MR | Zbl

[29] D. Yau, “The classical Hom–Yang–Baxter equation and Hom–Lie bialgebras”, Internat. Electron. J. Algebra, 17 (2015), 11–45 | DOI | MR | Zbl

[30] C. Boyallian, J. I. Liberati, “On pseudo-bialgebras”, J. Algebra, 372 (2012), 1–34 | DOI | MR | Zbl

[31] J. Liberati, “On conformal bialgebras”, J. Algebra, 319:6 (2008), 2295–2318 | DOI | MR | Zbl

[32] E. K. Sklyanin, L. A. Takhtadzhyan, L. D. Faddeev, “Kvantovyi metod obratnoi zadachi. I”, TMF, 40:2 (1979), 194–220 | DOI | MR

[33] L. A. Takhtadzhyan, L. D. Faddeev, “Kvantovyi metod obratnoi zadachi i $XYZ$ model Geizenberga”, UMN, 34:5(209) (1979), 13–63 | DOI | MR

[34] R. J. Baxter, “One dimensional anisotropic Heisenberg chain”, Ann. Phys., 70:2 (1972), 323–327 | DOI | MR

[35] C. N. Yang, “Some exact results for the many-body problem in one dimension with repulsive delta-function interaction”, Phys. Rev. Lett., 19:23 (1967), 1312–1315 | DOI | MR

[36] O. Babelon, C. M. Viallet, Integrable models, Yang–Baxter equations, and quantum groups. Part I, Trieste preprint SISSA 54 EP, Cambridge Univ. Press, Cambridge, 1989

[37] O. Babelon, C. M. Viallet, “Hamiltonian structures and Lax equations”, Phys. Lett. B, 237:3–4 (1990), 411–416 | DOI | MR

[38] I. M. Gelfand, I. V. Cherednik, “Abstraktnyi gamiltonov formalizm dlya klassicheskikh puchkov Yanga–Bakstera”, UMN, 38:3(231) (1983), 3–21 | DOI | MR | Zbl

[39] V. Kats, Beskonechnomernye algebry Li, Mir, M., 1993 | MR | MR

[40] C. M. Bai, L. Guo, X. Ni, “Nonabelian generalized Lax pairs, the classical Yang–Baxter equation and PostLie algebras”, Commun. Math. Phys., 297:2 (2010), 553–596 | DOI | MR | Zbl

[41] V. G. Drinfeld, “Kvantovye gruppy”, Differentsialnaya geometriya, gruppy Li i mekhanika. VIII, Zap. nauchn. sem. LOMI, 155, 1986, 18–49 | DOI | MR | Zbl

[42] Y. H. Sheng, C. M. Bai, “A new approach to Hom–Lie bialgebras”, J. Algebra, 399 (2014), 232–250 | DOI | MR | Zbl

[43] Q. X. Sun, “Generalization of $H$-pseudoalgebraic structures”, J. Math. Phys., 53:1 (2012), 012105, 18 pp. | DOI | MR | Zbl

[44] Q. X. Sun, “On $n$-ary Hom–Lie $H$-pseudoalgebras”, J. Phys. A: Math. Theor., 45:19 (2012), 195208, 13 pp. | DOI | MR | Zbl

[45] M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series, Benjamin, New York, 1969 | MR | Zbl