Cohomologies of relative Rota-Baxter Lie algebras with derivations and applications. (English) Zbl 1542.17033
Summary: The purpose of the present paper is to investigate cohomologies of relative Rota-Baxter Lie algebras with derivations and applications. First, we introduce a notion of relative Rota-Baxter LieDer pairs. Moreover, we construct cohomologies of relative Rota-Baxter LieDer pairs. Finally, we discuss abelian extensions, crossed modules and skeletal relative Rota-Baxter Lie 2-Der pairs in terms of cohomology groups.
MSC:
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
Keywords:
relative Rota-Baxter LieDer pair; cohomology; abelian extension; relative Rota-Baxter Lie 2-Der pair; crossed moduleReferences:
| [1] | Aguiar, M., On the associative analog of Lie bialgebras. J. Algebra, 492-532 (2001) · Zbl 0991.16033 |
| [2] | Ayala, V.; Kizil, E.; de Azevedo Tribuzy, I., On an algorithm for finding derivations of Lie algebras. Proyecciones, 81-90 (2012) · Zbl 1272.17021 |
| [3] | Ayala, V.; Tirao, J., Linear control systems on Lie groups and controllability, 47-64 · Zbl 0916.93015 |
| [4] | Baez, J. C.; Crans, A. S., Higher-dimensional algebra. VI. Lie 2-algebras. Theory Appl. Categ., 492-538 (2004) · Zbl 1057.17011 |
| [5] | Bai, C., A unified algebraic approach to the classical Yang-Baxter equation. J. Phys. A, 40, 11073-11082 (2007) · Zbl 1118.17008 |
| [6] | Baxter, G., An analytic problem whose solution follows from a simple algebraic identity. Pac. J. Math., 731-742 (1960) · Zbl 0095.12705 |
| [7] | Chtioui, T.; Mabrouk, S.; Makhlouf, A., Cohomology and deformations of \(\mathcal{O} \)-operators on Hom-associative algebras. J. Algebra, 727-759 (2022) · Zbl 1501.16007 |
| [8] | Coll, V. E.; Gerstenhaber, M.; Giaquinto, A., An explicit deformation formula with noncommuting derivations, 396-403 · Zbl 0684.16016 |
| [9] | Connes, A.; Kreimer, D., Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys., 249-273 (2000) · Zbl 1032.81026 |
| [10] | Das, A., Deformations of associative Rota-Baxter operators. J. Algebra, 144-180 (2020) · Zbl 1458.16008 |
| [11] | Das, A., Relative Rota-Baxter Leibniz algebras, their characterization and cohomology. Linear Multilinear Algebra (2022) |
| [12] | Das, A., Leibniz algebras with derivations. J. Homotopy Relat. Struct., 245-274 (2021) · Zbl 1477.17018 |
| [13] | Das, A.; Mandal, A., Extensions, deformations and categorifications of AssDer pairs |
| [14] | Das, A.; Mishra, S. K., Bimodules over relative Rota-Baxter algebras and cohomologies. Algebr. Represent. Theory (2022) |
| [15] | Das, A.; Misha, S. K., The \(L_\infty \)-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators · Zbl 1508.17024 |
| [16] | Guo, L.; Zhang, B., Renormalization of multiple zeta values. J. Algebra, 3770-3809 (2008) · Zbl 1165.11071 |
| [17] | Gu, N.; Guo, L., Generating functions from the viewpoint of Rota-Baxter algebras. Discrete Math., 536-554 (2015) · Zbl 1305.05010 |
| [18] | Hou, S.; Sheng, Y.; Zhou, Y., 3-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on 3-Lie algebras. J. Algebra, 103-129 (2023) · Zbl 1517.17005 |
| [19] | Jiang, J.; Sheng, Y., Representations and cohomologies of relative Rota-Baxter Lie algebras and applications. J. Algebra, 637-670 (2022) · Zbl 1500.17016 |
| [20] | Kupershmidt, A., What a classical \(r\)-matrix really is?. J. Nonlinear Math. Phys., 4, 448-488 (1999) · Zbl 1015.17015 |
| [21] | Lazarev, A.; Sheng, Y.; Tang, R., Deformations and homotopy theory of relative Rota-Baxter Lie algebras. Commun. Math. Phys., 1, 595-631 (2021) · Zbl 1476.17010 |
| [22] | Magid, A. R., Lectures on Differential Galois Theory. University Lecture Series (1994), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0855.12001 |
| [23] | Mishra, S. K.; Naolekar, A., \( \mathcal{O} \)-operators on Hom-Lie algebras. J. Math. Phys. (2020) |
| [24] | Rota, G. C., Baxter algebras and combinatorial identities I, II. Bull. Am. Math. Soc., 330-334 (1969) · Zbl 0319.05008 |
| [25] | Sheng, Y., Categorification of pre-Lie algebras and solutions of 2-graded classical Yang-Baxter equations. Theory Appl. Categ., 11, 269-294 (2019) · Zbl 1408.17009 |
| [26] | Sun, Q.; Wu, Z., Cohomologies of \(n\)-Lie algebras with derivations. Mathematics (2021) |
| [27] | Sun, Q.; Chen, S., Cohomologies and deformations of Lie triple systems with derivations. J. Algebra Appl. (2022) |
| [28] | Tang, R.; Bai, C.; Guo, L.; Sheng, Y., Deformations and their controlling cohomologies of \(\mathcal{O} \)-operators. Commun. Math. Phys., 2, 665-700 (2019) · Zbl 1440.17015 |
| [29] | Tang, R.; Frégier, Y.; Sheng, Y., Cohomologies of a Lie algebra with a derivation and applications. J. Algebra, 2, 65-99 (2019) · Zbl 1455.17020 |
| [30] | Uchino, K., Quantum analogy of Poisson geometry, related dendriform algebras and Rota-Baxter operators. Lett. Math. Phys., 91-109 (2008) · Zbl 1243.17002 |
| [31] | Voronov, T., Higher derived brackets and homotopy algebras. J. Pure Appl. Algebra, 133-153 (2005) · Zbl 1086.17012 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
