Bihom-pre-Lie superalgebras and related structures

Year 2022, , 1385 - 1402, 01.10.2022

Othmen Ncib

https://doi.org/10.15672/hujms.1035821

Abstract

Throughout this paper, we will study Rota-Baxter operators and super $\mathcal{O}$-operator of BiHom-associative superalgebras, BiHom-Lie superalgebras, BiHom-pre-Lie superalgebras and BiHom-$L$-dendriform superalgebras. Then we give some properties of BiHom-pre-Lie superalgebras constructed from BiHom-associative superalgebras, BiHom-Lie superalgebras and BiHom-$L$-dendriform superalgebras.

Keywords

BiHom-Lie superalgebras, BiHom-pre-Lie superalgebras, BiHom-L-dendriform superalgebras, representation, Rota-Baxter

References

• [1] E. Abdaoui, S. Mabrouk and A. Makhlouf, Rota-Baxter operators on Pre-Lie Superalgebras, Bull. Malays. Math. Sci. Soc. 42, 1567-1606, 2019.
• [2] M. Aguiar, Pre-Poisson algebras, Lett. Math. Phys. 54, 263–277, 2000.
• [3] M. Aguiar, Infinitesimal bialgebras, pre-Lie algebras and dendriform algebras, in "Hopf algebras", Lecture Notes in Pure and Appl. Math. 237, 1–33, 2004.
• [4] M. Aguiar and J. L. Loday, Quadri-algebras, J. Pure Appl. Algebra ,19(3), 205-221. 2004.
• [5] N. Aizawa and H. Sato, q-deformation of the Virasoro algebra with central extension, Phys. Lett. B, 256, 185-190, 1991.
• [6] F. Ammar and A. Makhlouf, Hom-Lie superalgebras and Hom-Lie admissible superalgebras, J. Algebra, 324, 1513–1528, 2010.
• [7] A. Andrada and S. Salamon, Complex product structure on Lie algebras, Forum Math. 17, 261–295, 2005.
• [8] F.V. Atkinson, Some aspects of Baxter’s functional equation, J. Math. Anal. Appl. 7, 1–30, 1967.
• [9] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10, 731-742, 1960.
• [10] C. M. Bai, A further study on non-abelian phase spaces: Left-symmetric algebraic approach and related geometry, Rev. Math. Phys. 18, 545–564, 2006.
• [11] C.M. Bai, A unified algebraic approach to classical Yang-Baxter equation, J. Phys. A, 40, 11073–11082, 2007.
• [12] C.M. Bai, O-operators of Loday algebras and analogues of the classical Yang-Baxter equation, Comm. Algebra, 38, 4277–4321, 2010.
• [13] C.M. Bai, L. Guo and X. Ni, Generalizations of the classical Yang-Baxter equation and O-operators, J. Math. Phys. 52, 063515, 2011.
• [14] C.M. Bai, L. Guo and X. Ni, O-operators on associative algebras and associative Yang-Baxter equations, Pacific J. Math. 256, 257–289, 2012.
• [15] C.M. Bai and L.G. Liu, Some results on L-dendriform algebras, J. Geom. Phys. 60, 940–950, 2010.
• [16] C.M. Bai and R. Zhang, On some left-symmetric superalgebras, J. Algebra Appl. 11 (5), 1250097, 2012.
• [17] I. Bakayoko, Hom-post-Lie modules, O-operators and some functors on Hom-algebras, arXiv preprint, arXiv:1610.02845, 2016.
• [18] A. Ben Hassine, S. Mabrouk and O. Ncib, 3-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras, Linear Multilinear Algebra, 70(1), 101-121, 2022.
• [19] M. Bordemann, Generalized Lax pairs, the modified classical Yang-Baxter equations, and affine geometry of Lie groups, Comm Math. Phys. 135(1), 201–216, 1990.
• [20] D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math. 4 (3), 323–357, 2006.
• [21] P. Cartier, On the structure of free Baxter algebras, Adv. Math. 9, 253–265, 1972.
• [22] F. Chapoton and M. Livernet, Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not. IMRN 8, 395–408, 2001.
• [23] Y. Cheng and H. Qi, Representations of BiHom-Lie algebras, Algebra Colloq. 29 (1), 125-142, 2022.
• [24] M. Chaichian, P. Kulish and J. Lukierski, q-deformed Jacobi identity, q-oscillators and q-deformed infinitedimensional algebras, Phys. Lett. B 237, 401-406, 1990.
• [25] B.Y. Chu, Symplectic homogeneous spaces, Trans. Amer. Math. Soc. 197, 145–159, 1974.
• [26] A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199, 203–242, 1998.
• [27] T.L. Curtright and C.K. Zachos, Deforming maps for quantum algebras, Phys. Lett. B, 243, 237–244, 1990.
• [28] J.M. Dardié and A. Medina, Algèbres de Lie Kahlériennes et double extension, J. Algebra, 185, 744–795, 1996.
• [29] J.M. Dardié and A. Medina, Double extension symplectique d’un groupe de Lie symplectique, Adv. Math. 117, 208–227, 1996.
• [30] A. Diatta and A. Medina, Classical Yang-Baxter equation and left-invariant affine geometry on Lie groups, Manuscipta. Math. 114, 477–486, 2004.
• [31] K. Ebrahimi-Fard, Loday-type algebras and the Rota-Baxter relation, Lett. Math. Phys. 61, 139–147, 2002.
• [32] K. Ebrahimi-Fard, On the associative Nijenhuis relation, Elect. J. Comb. 11(1), 2004.
• [33] K. Ebrahimi-Fard, D. Manchon and F. Patras, New identities in dendriform algebras, J. Algebra, 320, 708–727, 2008.
• [34] M. Gerstenhaber, The cohomology structure of associative ring, Ann. of Math. 78, 267–288, 1963.
• [35] M. Goze and E. Remm, Lie-admissible algebras and operads, J. Algebra, 273, 129– 152, 2004.
• [36] 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, 086, 2015.
• [37] M. Goncharov and V. Gubarev, Rota-Baxter operators of nonzero weight on the matrix algebra of order three, Linear Multilinear Algebra, 70 (6), 1055-1080, 2022.
• [38] L. Guo, An introduction to Rota-Baxter Algebra,Somerville: International Press 2(9), 2012.
• [39] J.-L. Koszul, Domaines bornés homogènes et orbites de groupes de transformations affines, Bull. Soc. Math. France, 89, 515–533, 1961.
• [40] B.A. Kupershmidt, Non-abelian phase spaces, J. Phys. A, 27, 2801–2809, 1994.
• [41] A. Lichnerowicz and A. Medina, On Lie group with left-invariant symplectic or Kählerian, Lett. Math. Phys. 16(3), 225–235, 1988.
• [42] Liu K.Q, Characterizations of the quantum Witt algebra, Lett. Math. Phys. 24, 257– 265, 1992.
• [43] J.-L. Loday, Dialgebras: In Dialgebras and related Operads, Springer, New York, 7–66, 2001.
• [44] J.B. Miller, Baxter operators and endomorphisms on Banach algebras, J. Math. Anal. Appl. 25, 503–520, 1969.
• [45] G.C. Rota, Baxter operators, In Gian-Carlo Rota on Combinatorics, Introductory Papes and commentaries, edited by Joseph. P.S. Kung, Birkhauser, Boston, 1995.
• [46] S. Wang and S. Guo, BiHom-Lie superalgebra structures, arXiv preprint arXiv:1610.02290, 2016.
• [47] E.A. Vasilieva and A.A. Mikhalev, Free left-symmetric superalgebras, Fundam. Prikl. Mat. 2, 611–613, 1996.
• [48] E.B. Vinberg, The theory of homogeneous cones, Tr. Mosk. Mat. Obs. 12, 303–358, 1963.