doi: 10.3934/era.2021059
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Constructions of three kinds of Bihom-superalgebras

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

* Corresponding author: chenly640@nenu.edu.cn

Received  May 2021 Revised  July 2021 Early access August 2021

Fund Project: Supported by the National Natural Science Foundation of China (Nos. 11771069 and 12071405)

The purpose of this paper is to study the constructions between Bihom-alternative superalgebras and Bihom-Malcev superalgebras and Bihom-Jordan superalgebras. First, we explain in detail that every regular Bihom-alternative superalgebra could be Bihom-Malcev-admissible superalgebra or Bihom-Jordan-admissible superalgebra. Next, the bimodules and $ T^*_\theta $-extensions of Bihom-alternative superalgebras are also discussed as properties of Bihom-alternative superalgebras.

Citation: Ying Hou, Liangyun Chen. Constructions of three kinds of Bihom-superalgebras. Electronic Research Archive, doi: 10.3934/era.2021059
References:
[1]

E. K. AbdaouiF. Ammar and A. Makhlouf, Hom-alternative, Hom-Malcev and Hom-Jordan superalgebras, Bull. Malays. Math. Sci. Soc., 40 (2017), 439-472.  doi: 10.1007/s40840-016-0323-5.  Google Scholar

[2]

N. Cantarini and V. G. Kac, Classification of linearly compact simple Jordan and generalized Poisson superalgebras, J. Algebra, 313 (2007), 100-124.  doi: 10.1016/j.jalgebra.2006.10.040.  Google Scholar

[3]

T. ChtiouiS. Mabrouk and A. Makhlouf, BiHom-alternative, BiHom-Malcev and BiHom-Jordan algebras, Rocky Mountain J. Math., 50 (2020), 69-90.  doi: 10.1216/rmj.2020.50.69.  Google Scholar

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T. ChtiouiS. Mabrouk and A. Makhlouf, BiHom-pre-alternative algebras and BiHom-alternative quadri-algebras, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 63 (2020), 3-21.   Google Scholar

[5]

A. Elduque and I. P. Shestakov, Irreducible non-Lie modules for Malcev superalgebras, J. Algebra, 173 (1995), 622-637.  doi: 10.1006/jabr.1995.1106.  Google Scholar

[6]

V. T. Filippov, On a varieties of Mal'tsev algebras, Algebra i Log., 20 (1981), 300-314.   Google Scholar

[7]

V. T. Filippov, Imbedding of Mal'tsev algebras into alternative algebras, Algebra i Log., 22 (1983), 443-465.   Google Scholar

[8]

A. Gohr, On hom-algebras with surjective twisting, J. Algebra, 324 (2010), 1483-1491.  doi: 10.1016/j.jalgebra.2010.05.003.  Google Scholar

[9]

G. Graziani, A. Makhlouf, C. Menini and F. Panaite, BiHom-associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras, SIGMA Symmetry Integrability Geom. Methods Appl., 11 (2015), Paper 086, 34 pp. doi: 10.3842/SIGMA.2015.086.  Google Scholar

[10]

S. GuoX. Zhang and S. Wang, The construction and deformation of BiHom-Novikov algebras, J. Geom. Phys., 132 (2018), 460-472.  doi: 10.1016/j.geomphys.2018.06.011.  Google Scholar

[11]

V. G. Kac, Classification of simple $\mathbb{Z}$-graded Lie superalgebras and simple Jordan superalgebras, Comm. Algebra, 5 (1977), 1375-1400.  doi: 10.1080/00927877708822224.  Google Scholar

[12]

Y. LiuL. Chen and Y. Ma, Hom-Nijienhuis operators and $T^*$-extensions of hom-Lie superalgebras, Linear Algebra Appl., 439 (2013), 2131-2144.  doi: 10.1016/j.laa.2013.06.006.  Google Scholar

[13]

L. LiuA. MakhloufC. Menini and F. Panaite, Rota-Baxter operators on BiHom-associative algebras and related structures, Colloq. Math., 161 (2020), 263-294.  doi: 10.4064/cm7877-5-2019.  Google Scholar

[14]

M. C. López-Díaz and I. P. Shestakov, Representations of exceptional simple alternative superalgebras of characteristic 3, Trans. Amer. Math. Soc., 354 (2002), 2745-2758.  doi: 10.1090/S0002-9947-02-02993-8.  Google Scholar

[15]

A. Makhlouf, Hom-alternative algebras and Hom-Jordan algebras, Int. Electron. J. Algebra, 8 (2010), 177-190.   Google Scholar

[16]

C. Martínez and E. Zelmanov, Representation theory of Jordan superalgebras. I, Trans. Amer. Math. Soc., 362 (2010), 815-846.  doi: 10.1090/S0002-9947-09-04883-1.  Google Scholar

[17]

J. NanC. Wang and Q. Zhang, Hom-Malcev superalgebras, Front. Math. China, 9 (2014), 567-584.  doi: 10.1007/s11464-014-0351-0.  Google Scholar

[18]

M. L. Racine and E. I. Zel'manov, Simple Jordan superalgebras with semisimple even part, J. Algebra, 270 (2003), 374-444.  doi: 10.1016/j.jalgebra.2003.06.012.  Google Scholar

[19]

Y. Sheng, Representations of hom-Lie algebras, Algebr. Represent. Theory, 15 (2012), 1081-1098.  doi: 10.1007/s10468-011-9280-8.  Google Scholar

[20]

I. P. Shestakov, Alternative and Jordan superalgebras, Siberian Adv. Math., 9 (1999), 83-99.   Google Scholar

[21]

I. P. Shestakov and A. Elduque, Prime non-Lie modules for Mal'tsev superalgebras, Algebra i Logika, 33 (1994), 253-262.  doi: 10.1007/BF00750852.  Google Scholar

[22]

I. P. Shestakov and S. Gonzalez, Non-associative algebra and its applications, Kluwer Academic Dordrecht, (1994), 372–378. Google Scholar

[23]

S. Wang and S. Guo, BiHom-Lie superalgebra structures and BiHom-Yang-Baxter equations, Adv. Appl. Clifford Algebr., 30 (2020), Paper No. 35, 18 pp. doi: 10.1007/s00006-020-01060-0.  Google Scholar

[24]

D. Yau, Hom-Maltsev, Hom-alternative, and Hom-Jordan algebras, Int. Electron. J. Algebra, 11 (2012), 177-217.   Google Scholar

[25]

E. I. Zel'manov and I. P. Shestakov, Prime alternative superalgebras and nilpotence of the radical of a free alternative algebra, Math. USSR-Izv., 37 (1991), 19–36.  Google Scholar

[26]

R. Zhang, D. Hou and C. Bai, A Hom-version of the affinizations of Balinskii-Novikov and Novikov superalgebras, J. Math. Phys., 52 (2011), 023505, 19 pp. doi: 10.1063/1.3546025.  Google Scholar

[27]

J. ZhaoL. Chen and L. Ma, Representations and $T^*$-extensions of hom-Jordan-Lie algebras, Comm. Algebra, 44 (2016), 2786-2812.  doi: 10.1080/00927872.2015.1065843.  Google Scholar

show all references

References:
[1]

E. K. AbdaouiF. Ammar and A. Makhlouf, Hom-alternative, Hom-Malcev and Hom-Jordan superalgebras, Bull. Malays. Math. Sci. Soc., 40 (2017), 439-472.  doi: 10.1007/s40840-016-0323-5.  Google Scholar

[2]

N. Cantarini and V. G. Kac, Classification of linearly compact simple Jordan and generalized Poisson superalgebras, J. Algebra, 313 (2007), 100-124.  doi: 10.1016/j.jalgebra.2006.10.040.  Google Scholar

[3]

T. ChtiouiS. Mabrouk and A. Makhlouf, BiHom-alternative, BiHom-Malcev and BiHom-Jordan algebras, Rocky Mountain J. Math., 50 (2020), 69-90.  doi: 10.1216/rmj.2020.50.69.  Google Scholar

[4]

T. ChtiouiS. Mabrouk and A. Makhlouf, BiHom-pre-alternative algebras and BiHom-alternative quadri-algebras, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 63 (2020), 3-21.   Google Scholar

[5]

A. Elduque and I. P. Shestakov, Irreducible non-Lie modules for Malcev superalgebras, J. Algebra, 173 (1995), 622-637.  doi: 10.1006/jabr.1995.1106.  Google Scholar

[6]

V. T. Filippov, On a varieties of Mal'tsev algebras, Algebra i Log., 20 (1981), 300-314.   Google Scholar

[7]

V. T. Filippov, Imbedding of Mal'tsev algebras into alternative algebras, Algebra i Log., 22 (1983), 443-465.   Google Scholar

[8]

A. Gohr, On hom-algebras with surjective twisting, J. Algebra, 324 (2010), 1483-1491.  doi: 10.1016/j.jalgebra.2010.05.003.  Google Scholar

[9]

G. Graziani, A. Makhlouf, C. Menini and F. Panaite, BiHom-associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras, SIGMA Symmetry Integrability Geom. Methods Appl., 11 (2015), Paper 086, 34 pp. doi: 10.3842/SIGMA.2015.086.  Google Scholar

[10]

S. GuoX. Zhang and S. Wang, The construction and deformation of BiHom-Novikov algebras, J. Geom. Phys., 132 (2018), 460-472.  doi: 10.1016/j.geomphys.2018.06.011.  Google Scholar

[11]

V. G. Kac, Classification of simple $\mathbb{Z}$-graded Lie superalgebras and simple Jordan superalgebras, Comm. Algebra, 5 (1977), 1375-1400.  doi: 10.1080/00927877708822224.  Google Scholar

[12]

Y. LiuL. Chen and Y. Ma, Hom-Nijienhuis operators and $T^*$-extensions of hom-Lie superalgebras, Linear Algebra Appl., 439 (2013), 2131-2144.  doi: 10.1016/j.laa.2013.06.006.  Google Scholar

[13]

L. LiuA. MakhloufC. Menini and F. Panaite, Rota-Baxter operators on BiHom-associative algebras and related structures, Colloq. Math., 161 (2020), 263-294.  doi: 10.4064/cm7877-5-2019.  Google Scholar

[14]

M. C. López-Díaz and I. P. Shestakov, Representations of exceptional simple alternative superalgebras of characteristic 3, Trans. Amer. Math. Soc., 354 (2002), 2745-2758.  doi: 10.1090/S0002-9947-02-02993-8.  Google Scholar

[15]

A. Makhlouf, Hom-alternative algebras and Hom-Jordan algebras, Int. Electron. J. Algebra, 8 (2010), 177-190.   Google Scholar

[16]

C. Martínez and E. Zelmanov, Representation theory of Jordan superalgebras. I, Trans. Amer. Math. Soc., 362 (2010), 815-846.  doi: 10.1090/S0002-9947-09-04883-1.  Google Scholar

[17]

J. NanC. Wang and Q. Zhang, Hom-Malcev superalgebras, Front. Math. China, 9 (2014), 567-584.  doi: 10.1007/s11464-014-0351-0.  Google Scholar

[18]

M. L. Racine and E. I. Zel'manov, Simple Jordan superalgebras with semisimple even part, J. Algebra, 270 (2003), 374-444.  doi: 10.1016/j.jalgebra.2003.06.012.  Google Scholar

[19]

Y. Sheng, Representations of hom-Lie algebras, Algebr. Represent. Theory, 15 (2012), 1081-1098.  doi: 10.1007/s10468-011-9280-8.  Google Scholar

[20]

I. P. Shestakov, Alternative and Jordan superalgebras, Siberian Adv. Math., 9 (1999), 83-99.   Google Scholar

[21]

I. P. Shestakov and A. Elduque, Prime non-Lie modules for Mal'tsev superalgebras, Algebra i Logika, 33 (1994), 253-262.  doi: 10.1007/BF00750852.  Google Scholar

[22]

I. P. Shestakov and S. Gonzalez, Non-associative algebra and its applications, Kluwer Academic Dordrecht, (1994), 372–378. Google Scholar

[23]

S. Wang and S. Guo, BiHom-Lie superalgebra structures and BiHom-Yang-Baxter equations, Adv. Appl. Clifford Algebr., 30 (2020), Paper No. 35, 18 pp. doi: 10.1007/s00006-020-01060-0.  Google Scholar

[24]

D. Yau, Hom-Maltsev, Hom-alternative, and Hom-Jordan algebras, Int. Electron. J. Algebra, 11 (2012), 177-217.   Google Scholar

[25]

E. I. Zel'manov and I. P. Shestakov, Prime alternative superalgebras and nilpotence of the radical of a free alternative algebra, Math. USSR-Izv., 37 (1991), 19–36.  Google Scholar

[26]

R. Zhang, D. Hou and C. Bai, A Hom-version of the affinizations of Balinskii-Novikov and Novikov superalgebras, J. Math. Phys., 52 (2011), 023505, 19 pp. doi: 10.1063/1.3546025.  Google Scholar

[27]

J. ZhaoL. Chen and L. Ma, Representations and $T^*$-extensions of hom-Jordan-Lie algebras, Comm. Algebra, 44 (2016), 2786-2812.  doi: 10.1080/00927872.2015.1065843.  Google Scholar

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