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Yıl 2022, , 199 - 217, 14.02.2022

Sami Mabrouk

https://doi.org/10.15672/hujms.923905
Cited By: 3

Öz

Kaynakça

  • [1] C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostMalcev algebras. Comm. Math. Phys. 297, 553-596, 2010.
  • [2] C. Bai, L. Guo and X. Ni, Generalizations of the classical Yang-Baxter equation and Kupershmidt operators. J. Math. Phys. 52, 063515, 2011.
  • [3] C. Bai, L. Guo and X. Ni, Relative Rota-Baxter operators and tridendriform algebras. J. Algebra Appl. 12, 5525-5537, 2013.
  • [4] D. Balavoine, Deformations of algebras over a quadratic Operads. Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math. 202, 207- 234, 1997.
  • [5] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10, 731-742, 1960.
  • [6] D. Burde, Simple left-symmetric algebras with solvable Lie algebra, Manuscripta Math. 95, 397-411, 1998.
  • [7] J. Carinena, J. Grabowski and G. Marmo Quantum bi-Hamiltonian systems. Internat. J. Modern Phys. A 15 (30), 4797-4810, 2000.
  • [8] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210, 249-273, 2000.
  • [9] I. Dorfman, Dirac structures and integrability of nonlinear evolution equations, Vol. 18. John Wiley & Son Limited, 1993.
  • [10] T.F. Fox, An introduction to algebraic deformation theory. J. Pure Appl. Algebra 84, 17-41, 1993.
  • [11] M. Gerstenhaber, On the deformation of rings and algebras. Ann. of Math. 79, 59-103, 1964.
  • [12] M. Gerstenhaber, On the deformation of rings and algebras. II. Ann. of Math. 84, 1-19, 1966.
  • [13] M. Gerstenhaber, On the deformation of rings and algebras. III. Ann. of Math. 88, 1-34, 1968.
  • [14] M. Gerstenhaber, On the deformation of rings and algebras. IV. Ann. of Math. 99, 257-276, 1974.
  • [15] L. Guo, An introduction to Rota-Baxter algebra. Surveys of modern mathematics, 4. International Press,Vol. 2. No.9. p.14. Somerville, MA; Higher Education Press, Beijing, 2012.
  • [16] F. Gürsey and C.H. Tze, On the role of division, Jordan and related algebras in particle physics, World Scientific, Singapore, 1996.
  • [17] F. S. Kerdman, Analytic Moufang loops in the large, Alg. Logic 18 , 325-347, 1980.
  • [18] M. Kontsevich and Y. Soibelman, Deformation theory. I [Draft], http://www.math.ksu.edu/ soibel/Book-vol1.ps, 2010.
  • [19] B. A. Kupershmidt, What a classical r-matrix really is, J. Nonlinear Math. Phy. 6, 448-488, 1999.
  • [20] E.N. Kuz’min, The connection between Mal’cev algebras and analytic Moufang loops, Alg. Logic 10, 1-14, 1971.
  • [21] J.L. Loday and B. Vallette, Algebraic operads, Springer, 2012.
  • [22] S. Madariaga, Splitting of operations for alternative and Malcev structures. Communications in Algebra, 45 (1), 183-197, 2014.
  • [23] A.I. Mal’tsev, Analytic loops, Mat. Sb. 36, 569-576, 1955.
  • [24] H.C. Myung, Malcev-admissible algebras, Progress in Math. Vol. 64, Birkhäuser, Boston, MA, 1986.
  • [25] P.T. Nagy, Moufang loops and Malcev algebras, Sem. Sophus Lie 3, 65-68, 1993.
  • [26] A. Nijenhuis and R. Richardson, Cohomology and deformations in graded Lie algebras. Bull. Amer. Math. Soc. 72, 1-29, 1966.
  • [27] A. Nijenhuis and R. W. Richardson, Deformation of Malcev algebra structures, J. Math. Mech. 17, 89-105, 1967.
  • [28] A. Nijenhuis and R. Richardson, Commutative algebra cohomology and deformations of Lie and associative algebras. J. Algebra 9, 42-105, 1968.
  • [29] S. Okubo, Introduction to octonions and other non-associative algebras in physics, Cambridge Univ. Press, Cambridge, UK, 1995.
  • [30] A. Sagle, Malcev algebras, Trans. Amer. Math. Soc. 101, 426-458, 1961.
  • [31] L.V. Sabinin, Smooth quasigroups and loops, Kluwer Academic, The Netherlands, 1999.
  • [32] K. Uchino, Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators, Lett. Math. Phys. 85, 91-109, 2008.
  • [33] K. Uchino, Twisting on associative algebras and Rota-Baxter type operators. J. Noncommut. Geom. 4, 349-379, 2010.

Deformation of Kupershmidt operators and Kupershmidt-Nijenhuis structures of a Malcev algebra

Yıl 2022, , 199 - 217, 14.02.2022

Sami Mabrouk

https://doi.org/10.15672/hujms.923905
Cited By: 3

Öz

The aim of this paper is to study infinitesimal deformations of a Malcev algebra with a representation and introduce the notion of Nijenhuis pair, which gives a trivial deformation of a Malcev algebra with a representation. We introduce the notion of Kupershmidt-(dual-)Nijenhuis structure on a Malcev algebra with a representation. Furthermore, we show that a Kupershmidt-(dual-)Nijenhuis structure gives rise to a hierarchy of Kupershmidt operators. Finally, we establish a deformation theory of Kupershmidt operators in consistence with the general principles of deformation theories and introduce the notion of Nijenhuis elements.

Anahtar Kelimeler

altenative algebras Malcev algebras representations (dual-)Nijenhuispair Kupershmidt operator Kupershmidt-(dual-)Nijenhuis structure

Kaynakça

  • [1] C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostMalcev algebras. Comm. Math. Phys. 297, 553-596, 2010.
  • [2] C. Bai, L. Guo and X. Ni, Generalizations of the classical Yang-Baxter equation and Kupershmidt operators. J. Math. Phys. 52, 063515, 2011.
  • [3] C. Bai, L. Guo and X. Ni, Relative Rota-Baxter operators and tridendriform algebras. J. Algebra Appl. 12, 5525-5537, 2013.
  • [4] D. Balavoine, Deformations of algebras over a quadratic Operads. Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math. 202, 207- 234, 1997.
  • [5] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10, 731-742, 1960.
  • [6] D. Burde, Simple left-symmetric algebras with solvable Lie algebra, Manuscripta Math. 95, 397-411, 1998.
  • [7] J. Carinena, J. Grabowski and G. Marmo Quantum bi-Hamiltonian systems. Internat. J. Modern Phys. A 15 (30), 4797-4810, 2000.
  • [8] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210, 249-273, 2000.
  • [9] I. Dorfman, Dirac structures and integrability of nonlinear evolution equations, Vol. 18. John Wiley & Son Limited, 1993.
  • [10] T.F. Fox, An introduction to algebraic deformation theory. J. Pure Appl. Algebra 84, 17-41, 1993.
  • [11] M. Gerstenhaber, On the deformation of rings and algebras. Ann. of Math. 79, 59-103, 1964.
  • [12] M. Gerstenhaber, On the deformation of rings and algebras. II. Ann. of Math. 84, 1-19, 1966.
  • [13] M. Gerstenhaber, On the deformation of rings and algebras. III. Ann. of Math. 88, 1-34, 1968.
  • [14] M. Gerstenhaber, On the deformation of rings and algebras. IV. Ann. of Math. 99, 257-276, 1974.
  • [15] L. Guo, An introduction to Rota-Baxter algebra. Surveys of modern mathematics, 4. International Press,Vol. 2. No.9. p.14. Somerville, MA; Higher Education Press, Beijing, 2012.
  • [16] F. Gürsey and C.H. Tze, On the role of division, Jordan and related algebras in particle physics, World Scientific, Singapore, 1996.
  • [17] F. S. Kerdman, Analytic Moufang loops in the large, Alg. Logic 18 , 325-347, 1980.
  • [18] M. Kontsevich and Y. Soibelman, Deformation theory. I [Draft], http://www.math.ksu.edu/ soibel/Book-vol1.ps, 2010.
  • [19] B. A. Kupershmidt, What a classical r-matrix really is, J. Nonlinear Math. Phy. 6, 448-488, 1999.
  • [20] E.N. Kuz’min, The connection between Mal’cev algebras and analytic Moufang loops, Alg. Logic 10, 1-14, 1971.
  • [21] J.L. Loday and B. Vallette, Algebraic operads, Springer, 2012.
  • [22] S. Madariaga, Splitting of operations for alternative and Malcev structures. Communications in Algebra, 45 (1), 183-197, 2014.
  • [23] A.I. Mal’tsev, Analytic loops, Mat. Sb. 36, 569-576, 1955.
  • [24] H.C. Myung, Malcev-admissible algebras, Progress in Math. Vol. 64, Birkhäuser, Boston, MA, 1986.
  • [25] P.T. Nagy, Moufang loops and Malcev algebras, Sem. Sophus Lie 3, 65-68, 1993.
  • [26] A. Nijenhuis and R. Richardson, Cohomology and deformations in graded Lie algebras. Bull. Amer. Math. Soc. 72, 1-29, 1966.
  • [27] A. Nijenhuis and R. W. Richardson, Deformation of Malcev algebra structures, J. Math. Mech. 17, 89-105, 1967.
  • [28] A. Nijenhuis and R. Richardson, Commutative algebra cohomology and deformations of Lie and associative algebras. J. Algebra 9, 42-105, 1968.
  • [29] S. Okubo, Introduction to octonions and other non-associative algebras in physics, Cambridge Univ. Press, Cambridge, UK, 1995.
  • [30] A. Sagle, Malcev algebras, Trans. Amer. Math. Soc. 101, 426-458, 1961.
  • [31] L.V. Sabinin, Smooth quasigroups and loops, Kluwer Academic, The Netherlands, 1999.
  • [32] K. Uchino, Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators, Lett. Math. Phys. 85, 91-109, 2008.
  • [33] K. Uchino, Twisting on associative algebras and Rota-Baxter type operators. J. Noncommut. Geom. 4, 349-379, 2010.
Toplam 33 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Sami Mabrouk 0000-0003-2610-3262

Yayımlanma Tarihi 14 Şubat 2022
Yayımlandığı Sayı Yıl 2022

Kaynak Göster

APA Mabrouk, S. (2022). Deformation of Kupershmidt operators and Kupershmidt-Nijenhuis structures of a Malcev algebra. Hacettepe Journal of Mathematics and Statistics, 51(1), 199-217. https://doi.org/10.15672/hujms.923905
AMA Mabrouk S. Deformation of Kupershmidt operators and Kupershmidt-Nijenhuis structures of a Malcev algebra. Hacettepe Journal of Mathematics and Statistics. Şubat 2022;51(1):199-217. doi:10.15672/hujms.923905
Chicago Mabrouk, Sami. “Deformation of Kupershmidt Operators and Kupershmidt-Nijenhuis Structures of a Malcev Algebra”. Hacettepe Journal of Mathematics and Statistics 51, sy. 1 (Şubat 2022): 199-217. https://doi.org/10.15672/hujms.923905.
EndNote Mabrouk S (01 Şubat 2022) Deformation of Kupershmidt operators and Kupershmidt-Nijenhuis structures of a Malcev algebra. Hacettepe Journal of Mathematics and Statistics 51 1 199–217.
IEEE S. Mabrouk, “Deformation of Kupershmidt operators and Kupershmidt-Nijenhuis structures of a Malcev algebra”, Hacettepe Journal of Mathematics and Statistics, c. 51, sy. 1, ss. 199–217, 2022, doi: 10.15672/hujms.923905.
ISNAD Mabrouk, Sami. “Deformation of Kupershmidt Operators and Kupershmidt-Nijenhuis Structures of a Malcev Algebra”. Hacettepe Journal of Mathematics and Statistics 51/1 (Şubat 2022), 199-217. https://doi.org/10.15672/hujms.923905.
JAMA Mabrouk S. Deformation of Kupershmidt operators and Kupershmidt-Nijenhuis structures of a Malcev algebra. Hacettepe Journal of Mathematics and Statistics. 2022;51:199–217.
MLA Mabrouk, Sami. “Deformation of Kupershmidt Operators and Kupershmidt-Nijenhuis Structures of a Malcev Algebra”. Hacettepe Journal of Mathematics and Statistics, c. 51, sy. 1, 2022, ss. 199-17, doi:10.15672/hujms.923905.
Vancouver Mabrouk S. Deformation of Kupershmidt operators and Kupershmidt-Nijenhuis structures of a Malcev algebra. Hacettepe Journal of Mathematics and Statistics. 2022;51(1):199-217.

Cited By

Super Kupershmidt Operators on Pre-malcev Superalgebras
Frontiers of Mathematics
https://doi.org/10.1007/s11464-023-0045-6
Kupershmidt operators on Hom-Malcev algebras and their deformation
International Journal of Geometric Methods in Modern Physics
https://doi.org/10.1142/S0219887823500469
Super Kupershmidt operators and Kupershmidt-Nijenhuis structures on Malcev superalgebras
Communications in Algebra
https://doi.org/10.1080/00927872.2023.2215870

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