Biderivations and strong commutativity-preserving maps on parabolic subalgebras of simple Lie algebras

References

• Brešar M. Commuting maps: a survey. Taiwanese J Math. 2004;8(3):361–397. doi: 10.11650/twjm/1500407660
   Web of Science ®Google Scholar
• Brešar M, Zhao K. Biderivations and commuting linear maps on Lie algebras. J Lie Thoery. 2018;28(3):885–900.
   Web of Science ®Google Scholar
• Chen Z. Biderivations and linear commuting maps on simple generalized Witt algebras over a field. Electron J Linear Algebra. 2016;31(1):1–12. doi: 10.13001/1081-3810.3100
   Web of Science ®Google Scholar
• Cheng X, Wang M, Sun J, et al. Biderivations and linear commuting maps on the Lie algebra gca. Linear Multilinear Algebra. 2017;65(12):2483–2493. doi: 10.1080/03081087.2016.1277688
   Web of Science ®Google Scholar
• Han X, Wang D, Xia C. Linear commuting maps and biderivations on the Lie algebras W(a,b). J Lie Thoery. 2016;26:777–786.
   Web of Science ®Google Scholar
• Wang D, Yu X. Biderivations and linear commuting maps on the Schrödinger-Virasoro Lie algebras. Comm Algebra. 2013;41(6):2166–2173. doi: 10.1080/00927872.2012.654551
   Web of Science ®Google Scholar
• Wang D, Yu X, Chen Z. Biderivations of the parabolic subalgebras of simple Lie algebras. Comm Algebra. 2011;39(11):4097–4104. doi: 10.1080/00927872.2010.517820
   Web of Science ®Google Scholar
• Tang X. Biderivations of finite dimensional complex simple Lie algebras. Linear Multilinear Algebra. 2018;66(2):250–259. doi: 10.1080/03081087.2017.1295433
   Web of Science ®Google Scholar
• Liu X, Guo X, Zhao K. Biderivations of block Lie algebras. Linear Algebra Appl. 2018;538:43–55. doi: 10.1016/j.laa.2017.10.011
   Web of Science ®Google Scholar
• Tang X. Biderivations and commutative post-Lie algebra structures on the Lie algebra W(a,b). Taiwanese J Math. 2018;22(6):1347–1366. doi: 10.11650/tjm/180305
   Web of Science ®Google Scholar
• Wang D, Zhang W, Chen Z. Product zero derivations of the parabolic subalgebras of simple Lie algebras. J Lie Theory. 2010;20:167–174.
   Web of Science ®Google Scholar
• Benkovič D, Eremita D. Commuting traces and commutativity preserving maps on triangular algebras. J Algebra. 2004;280:797–824. doi: 10.1016/j.jalgebra.2004.06.019
   Web of Science ®Google Scholar
• Brešar M, Šemrl P. Commutativity preserving linear maps on central simple algebras. J Algebra. 2005;284:102–110. doi: 10.1016/j.jalgebra.2004.09.014
   Web of Science ®Google Scholar
• Chan GH, Lim MH. Linear transformations on symmetric matrices that preserve commutativity. Linear Algebra Appl. 1982;47:11–22. doi: 10.1016/0024-3795(82)90222-1
   Web of Science ®Google Scholar
• Choi MD, Jafarian A, Radjavi H. Linear maps preserving commutativity. Linear Algebra Appl. 1987;87:227–242. doi: 10.1016/0024-3795(87)90169-8
   Web of Science ®Google Scholar
• Marcoux LW, Sourour AR. Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras. Linear Algebra Appl. 1999;288:89–104. doi: 10.1016/S0024-3795(98)10182-9
   Web of Science ®Google Scholar
• Wang D. Invertible linear maps on Borel subalgebras preserving zero Lie products. J Algebra. 2014;406(5):321–336. doi: 10.1016/j.jalgebra.2014.02.023
   Google Scholar
• Wang D, Chen Z. Invertible linear maps on simple Lie algebras preserving commutativity. Proc Amer Math Soc. 2011;139:3881–3893. doi: 10.1090/S0002-9939-2011-10834-7
   Web of Science ®Google Scholar
• Bell HE, Daif MN. On commutativity and strong commutativity preserving maps. Canad Math Bull. 1994;37:443–447. doi: 10.4153/CMB-1994-064-x
   Web of Science ®Google Scholar
• Brešar M, Miers CR. Strong commutativity preserving maps of semiprime rings. Canad Math Bull. 1994;37:457–460. doi: 10.4153/CMB-1994-066-4
   Web of Science ®Google Scholar
• Deng Q, Ashraf M. On strong commutativity preserving maps. Results Math. 1996;30:259–263. doi: 10.1007/BF03322194
   Google Scholar
• Filippis VD, Scudo G. Strong commutativity and Engel condition preserving maps in prime and semiprime rings. Linear Multilinear Algebra. 2013;61:917–938. doi: 10.1080/03081087.2012.716433
   Web of Science ®Google Scholar
• Lin J, Liu C. Strong commutativity preserving maps in prime rings with involution. Linear Algebra Appl. 2010;432:14–23. doi: 10.1016/j.laa.2009.06.036
   Web of Science ®Google Scholar
• Lin J, Liu C. Strong commutativity preserving maps on Lie ideals. Linear Algebra Appl. 2008;428:1601–1609. doi: 10.1016/j.laa.2007.10.006
   Web of Science ®Google Scholar
• Liu C. Strong commutativity preserving generalized derivations on right ideals. Monatsh Math. 2012;166:453–465. doi: 10.1007/s00605-010-0281-1
   Web of Science ®Google Scholar
• Ma J, Xu X, Niu F. Strong commutativity preserving generalized derivations on semiprime rings. Acta Math Sin (Engl Ser). 2008;24:1835–1842. doi: 10.1007/s10114-008-7445-0
   Web of Science ®Google Scholar
• Wang D, Chen Z. Quasi-automorphisms of Lie algebras. Comm Algebra. 2011;39(7):2388–2395. doi: 10.1080/00927872.2010.488682
   Web of Science ®Google Scholar
• Burde D, Dekimpe K. Post-Lie algebra structures on pairs of Lie algebras. J Algebra. 2016;464:226–245. doi: 10.1016/j.jalgebra.2016.05.026
   Web of Science ®Google Scholar
• Humphreys JE. Introduction to Lie algebras and representation theory. New York: Springer; 1972.
   Google Scholar
• Meng D. Some results on complete Lie algebras. Comm Algebra. 1994;22(13):5457–5507. doi: 10.1080/00927879408825141
   Web of Science ®Google Scholar
• Chen Z, Wang D. Non-linear maps satisfying derivability on standard parabolic subalgebras of finite-dimensional simple Lie algebras. Linear Multilinear Algebra. 2011;59(3):261–270. doi: 10.1080/03081080903402921
   Web of Science ®Google Scholar