The Concept of Parafree Zinbiel Algebras

Abstract

Pf (parafree) Zinbiel (PfZin) algebras, a generalization of Leibniz algebras, share various traits with free Zinbiel algebras. This article delves into the intricacies of PfZin algebras, presenting their structure and exploring significant findings analogous to those in parafree Leibniz algebras. The focus extends to properties of subalgebras and quotient algebras within the realm of PfZin algebras. Additionally, the direct sum of these algebras is examined, demonstrating that the amalgamation of two PfZin algebras yields a Zinbiel algebra. A new connection between weak Hopf algebras and PfZin algeras constructed. Moreover, from the direct sum of PfZin algebras weak Hopf algebra is handled and construction of weak Hopf algebra usuing PfZin algebra is showed.

Keywords

• Zinbiel algebra
• Subalgebras
• Division algebras
• Direct sum
• Weak Hopf algebra

References

1. [1]. Aissaoui, R., Makhlouf, A., & Silvestrov, S. (2014). Hom-Lie coalgebras, Hom-Zinbiel algebras and Hom-Hopf algebras. Frontiers in Mathematics, 1(1), 89-111.
2. [2]. Bahturin, Y. I . 1987. Density relations in Lie algebras, VNU Science Press, Utrecht.
3. [3]. Baur, H. 1980. A note on parafree Lie algebras, Commun. in Alg.; 8(10): 953-960.
4. [4]. Baur, H. 1978. Parafreie Lie algebren and homologie, Diss. Eth Nr.; 6126: 60 pp.
5. [5]. Bloh, A.M. 1965. A generalization of the concept of Lie algebra, Dokl. Akad. Nauk SSSR; 165: 471-473.
6. [6]. Bloh, A.M. 1971. A certain generalization of the concept of Lie algebra, Algebra and Number Theory, Moskow. Gos. Ped. Inst. U`cen; 375: 9-20.
7. [7]. Böhm, Gabriella; Nill, Florian; Szlachányi, Kornel (1999). "Weak Hopf algebras. I. Integral theory and C-structure". Journal of Algebra. 221 (2): 385–438. doi:10.1006/jabr.1999.7984.
8. [8]. Elduque, A., & Makhlouf, A. 2020. Super Zinbiel algebras. Symmetry, Integrability and Geometry: Methods and Applications, 16, 017.

9. [9]. Ekici, N, Velioğlu, Z. 2014. Unions of Parafree Lie algebras, Algebra; Article ID 385397.
10. [10]. Ekici N, Velioğlu Z. 2015. Direct Limit of Parafree Lie algebras, Journal of Lie Theory; 25(2): 477-484.
11. [11]. Evans, T. 1969. Finitely presented loops, lattices, etc. are Hopfian, J. London Math. Soc.; 44: 551-552.
12. [12]. Gunzburg, V. & Kapranov, M. 1994. Koszul duality for operads, Duke Math. J. 76, 203-273.
13. [13]. Loday, J. L. 1995. Cup-product for Leibniz cohomology and dual Leibniz algebras, Math Scand. 77 (2), 189-196.
14. [14]. Loday, J. L. 2001. Dialgebras, In J.L. Loday, F. Chapoton, A. Frabetti and F. Goichot: Dialgebras and Related Operads. Lecture Notes in Mathematics, Springer, Berlin, Heidelberg vol 1763.
15. [15]. Loday, J. L., Pirashvili, T. 1993. Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann.; 269(1): 139-158.
16. [16]. Makhlouf, A., & Goze, M. 2010. Zinbiel algebras and the Wajnryb homomorphism. Journal of Algebra, 324(3), 874-890.
17. [17]. Mansuroğlu, N. 2022. On parafree Leibniz algebras. Celal Bayar University of Science, 18,3, 275-278.
18. [18]. Poland, A. 2015. Zinbiel Algebras: From Foundations to Applications. Springer.
19. [19]. Silvestrov, S., & Makhlouf, A. 2014. Deformations of Zinbiel algebras and higher homotopy algebras. Journal of Noncommutative Geometry, 8(2), 421-47.
20. [20]. Velioğlu, Z. 2013. Subalgebras and Quotient algebras of Parafree Lie algebras, I. Journal Pure and Applied Maths.; 83(3) 507-514.