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GRADED POST-LIE ALGEBRA STRUCTURES, ROTA-BAXTER OPERATORS AND YANG-BAXTER EQUATIONS ON THE W-ALGEBRA W(2, 2)

  • Tang, Xiaomin (School of Mathematical Science Heilongjiang University) ;
  • Zhong, Yongyue (School of Mathematical Science Heilongjiang University)
  • Received : 2017.11.21
  • Accepted : 2018.09.07
  • Published : 2018.11.30

Abstract

In this paper, we characterize the graded post-Lie algebra structures on the W-algebra W(2, 2). Furthermore, as applications, the homogeneous Rota-Baxter operators on W(2, 2) and solutions of the formal classical Yang-Baxter equation on $W(2,2){\ltimes}_{ad^*} W(2,2)^*$ are studied.

Acknowledgement

Supported by : NNSFC, NSF of Heilongjiang Province, Heilongjiang University

References

  1. C. Bai, L. Guo, and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras, Comm. Math. Phys. 297 (2010), no. 2, 553-596. https://doi.org/10.1007/s00220-010-0998-7
  2. G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10 (1960), 731-742. https://doi.org/10.2140/pjm.1960.10.731
  3. R. J. Baxter, Partition function of the eight-vertex lattice model, Ann. Physics 70 (1972), 193-228. https://doi.org/10.1016/0003-4916(72)90335-1
  4. D. Burde and K. Dekimpe, Post-Lie algebra structures on pairs of Lie algebras, J. Algebra 464 (2016), 226-245. https://doi.org/10.1016/j.jalgebra.2016.05.026
  5. D. Burde, K. Dekimpe, and K. Vercammen, Affine actions on Lie groups and post-Lie algebra structures, Linear Algebra Appl. 437 (2012), no. 5, 1250-1263. https://doi.org/10.1016/j.laa.2012.04.007
  6. D. Burde and W. A. Moens, Commutative post-Lie algebra structures on Lie algebras, J. Algebra 467 (2016), 183-201. https://doi.org/10.1016/j.jalgebra.2016.07.030
  7. H. Chen and J. Li, Left-symmetric algebra structures on the W-algebra W(2, 2), Linear Algebra Appl. 437 (2012), no. 7, 1821-1834. https://doi.org/10.1016/j.laa.2012.05.010
  8. C. Chu and L. Guo, Localization of Rota-Baxter algebras, J. Pure Appl. Algebra 218 (2014), no. 2, 237-251. https://doi.org/10.1016/j.jpaa.2013.05.009
  9. K. Ebrahimi-Fard, A Lundervold, I Mencattini, and H. Z Munthe-Kaas, Post-Lie algebras and isospectral flows, SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 093, 16 pp.
  10. K. Ebrahimi-Fard, A. Lundervold, and H. Munthe-Kaas, On the Lie enveloping algebra of a post-Lie algebra, J. Lie Theory 25 (2015), no. 4, 1139-1165.
  11. K. Ebrahimi-Fard, I. Mencattini, and H. Munthe-Kaas, Post-Lie algebras and factorization theorems, J. Geom. Phys. 119 (2017), 19-33. https://doi.org/10.1016/j.geomphys.2017.04.007
  12. S. L. Gao, C. P. Jiang, and Y. F. Pei, Derivations, central extensions and automorphisms of a Lie algebra, Acta Math. Sinica (Chin. Ser.) 52 (2009), no. 2, 281-288.
  13. X. Gao, M. Liu, C Bai, and N. Jing, Rota-Baxter operators on Witt and Virasoro algebras, J. Geom. Phys. 108 (2016), 1-20. https://doi.org/10.1016/j.geomphys.2016.06.007
  14. L. Guo, An Introduction to Rota-Baxter Algebra, Surveys of Modern Mathematics, 4, International Press, Somerville, MA, 2012.
  15. W. Jiang and W. Zhang, Verma modules over the W(2, 2) algebras, J. Geom. Phys. 98 (2015), 118-127. https://doi.org/10.1016/j.geomphys.2015.07.029
  16. R. Kashaev, The Yang-Baxter relation and gauge invariance, J. Phys. A 49 (2016), no. 16, 164001, 16 pp. https://doi.org/10.1088/1751-8113/49/16/164001
  17. H. Z. Munthe-Kaas and A. Lundervold, On post-Lie algebras, Lie-Butcher series and moving frames, Found. Comput. Math. 13 (2013), no. 4, 583-613. https://doi.org/10.1007/s10208-013-9167-7
  18. Y. Pan, Q. Liu, C. Bai, and L. Guo, PostLie algebra structures on the Lie algebra SL(2, ${\mathbb{C}}$), Electron. J. Linear Algebra 23 (2012), 180-197.
  19. G. Radobolja, Subsingular vectors in Verma modules, and tensor product of weight modules over the twisted Heisenberg-Virasoro algebra and W(2, 2) algebra, J. Math. Phys. 54 (2013), no. 7, 071701, 24 pp. https://doi.org/10.1063/1.4813439
  20. G.-C. Rota, Baxter operators, an introduction, in Gian-Carlo Rota on combinatorics, 504-512, Contemp. Mathematicians, Birkhauser Boston, Boston, MA, 1995.
  21. X. Tang, Post-Lie algebra structures on the Witt algebra, arXiv:1701.00200, 2017.
  22. X. Tang, Biderivations, linear commuting maps and commutative post-Lie algebra structures on W-algebras, Comm. Algebra 45 (2017), no. 12, 5252-5261. https://doi.org/10.1080/00927872.2017.1302456
  23. X. Tang and Y. Zhang, Post-Lie algebra structures on solvable Lie algebra t(2, ${\mathbb{C}}$), Linear Algebra Appl. 462 (2014), 59-87. https://doi.org/10.1016/j.laa.2014.08.019
  24. X. Tang, Y. Zhang, and Q. Sun, Rota-Baxter operators on 4-dimensional complex simple associative algebras, Appl. Math. Comput. 229 (2014), 173-186.
  25. B. Vallette, Homology of generalized partition posets, J. Pure Appl. Algebra 208 (2007), no. 2, 699-725. https://doi.org/10.1016/j.jpaa.2006.03.012
  26. Y. Wang, Q. Geng, and Z. Chen, The superalgebra of W(2, 2) and its modules of the intermediate series, Comm. Algebra 45 (2017), no. 2, 749-763. https://doi.org/10.1080/00927872.2016.1175450
  27. C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312-1315. https://doi.org/10.1103/PhysRevLett.19.1312
  28. W. Zhang and C. Dong, W-algebra W(2, 2) and the vertex operator algebra $L({\frac{1}{2}},0){\otimes}L({\frac{1}{2}},0)$, Comm. Math. Phys. 285 (2009), no. 3, 991-1004. https://doi.org/10.1007/s00220-008-0562-x