A Soft View on Lie Groups and Lie Hypergroups

Abstract

The aim of this paper is to conclude and investigate the concepts of soft Lie groups over a Lie group and soft Lie hypergroups over a Lie hypergroup. Some important soft Lie constructions are suggested, several results and examples are obtained. Morever, two new categories of the soft Lie groups and soft Lie hypergroups are presented, and substructures of these concepts are studied in detail. Finally, relations between different soft algebraic structures of the soft Lie groups and soft Lie hypergroups are examined.

Keywords

• Lie group
• Lie hypergroup
• Soft set
• Soft lie group
• Soft lie hypergroup

References

1. [1] Akram M. Bipolar fuzzy soft Lie algebras. Quasigroups and related systems 2013; 21(1): 1–10.
2. [2] Akram M, Feng F. Soft intersection Lie algebras. Quasigroups and Related Systems 2013; 21(1): 11–18.
3. [3] Aktas H, Cagman N. Soft sets and soft groups. Information Sciences 2007; 77(13): 2726–2735. https://doi.org/10.1016/j.ins.2006.12.008
4. [4] Amini M, Heydari A, Toomanian M. Lie hypergroups. Journal of Lie Theory 2013; 23(1): 127–142. [5] Belinfante JGF, Kolman B. A survey of Lie groups and Lie algebras with applications and computational methods. Second Edition, 1989.
5. [6] Coelho P, Nunes U. Lie algebra application to mobile robot control : a tutorial. Robotica 2003; 21(5): 483 – 493.
6. [7] Cagman N, Citak F, Aktas H. Soft int-group and its applications to group theory. Neural Computing and Applications 2012; 21(1) : 151–158.10. https://doi.org/10.1007/s00521-011-0752-x
7. [8] Cagman N, Enginoglu S. Soft set theory and uniint decision making. European journal of operational research 2010; 207(2): 848–855. https://doi.org/10.1016/j.ejor.2010.05.004
8. [9] Estaji AA, Eghdami H, Haghdadi T. Category of soft Lie algebra. Soft Computing 2020; 24(5): 3067–3076. https://doi.org/10.1007/s00500-019-04583-2.

9. [10] Gilmore R. Lie groups, Lie algebras, and some of their applications. Courier Corporation, 2006.
10. [11] Carinena, J. F., Ramos, A. Applications of Lie systems in quantum mechanics and control theory. arXiv preprint math-ph/0305021, 2003.
11. [12] Kazanci O, Yilmaz S, Yamak S. Soft sets and soft BCH-algebras. Hacettepe Journal of Mathematics and Statistics 2010; 39(2) : 205–217.
12. [13] Maji PK, Biswas R, Roy AR. Soft set theory, Computers and Mathematics with Applications 2003; 45(4-5): 555–562. https://doi.org/10.1016/S0898- 1221(03)00016-6.
13. [14] Molodtsov DA. Soft set theory-First results. Computers and Mathematics with Applications 1999; 37(4-5): 19–31.
14. [15] Nazmul S, Samanta SK. Soft topological groups, Kochi Journal of Mathematics 2010; 5: 151–161.
15. [16] Oguz G, Icen I, Gursoy MH. Actions of soft groups. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics 2019; 68(1): 1163–1174.
16. [17] Oguz G, Gursoy MH, Icen I. On soft topological categories. Hacettepe Journal of Mathematics and Statistics 2019; 48(6): 1675–1681. https://doi.org/10.15672/HJMS.2018.600
17. [18] Oguz G. Soft topological transformation groups. Mathematics 2020; 8(9) : 1545. https://doi.org/10.3390/math8091545
18. [19] Oguz G. On soft topological hypergroups. Journal of Hyperstructures 2020; 9(2): 81–95.
19. [20] Shabir M, Naz M. On soft topological spaces. Computers and Mathematics with Applications 2011; 61(7): 1786–1799. https://doi.org/10.1016/j.camwa.2011.02.006.
20. [21] Oguz G, Icen I, Gursoy M.H, Lie Rough Groups, Filomat,32:16 (2018), 57355741
21. [22] Toomanian M, Amini M, Heydari A. Representations of double coset Lie hypergoups, Iranian Journal of Mathematical Sciences and Informatics 2016; 11: 87–96. https://doi.org/10.7508/ijmsi.2016.02.006.
22. [23] Oguz G, Icen I, Gursoy M.H, A new concept in the soft theory: soft groupoids, Southeast Asian Bulletin of Mathematics 44 (4), 555-565
23. [24] Zou Y, Xiao Z. Data analysis approaches of soft sets under incomplete information. Knowledge-Based Systems 2008; 21(8): 941–945. https://doi.org/10.1016/j.knosys.2008.04.004.
24. [25] Oguz G, Gursoy M.H, Icen I, A soft approach to ring-groupoids,ITM Web of Conferences (Vol. 22, p. 01012). EDP Sciences, 2018.