Let R be a commutative ring with unity and let 𝒰 be a unital algebra over ℛ (or a field 𝔽). An R-linear map L : 𝒰 → 𝒰 is called a Lie derivation on 𝒰 if L([u, v]) = [L(u), v] + [u,L(v)] holds for all u, v 𝜖 𝒰. For scalar 𝜉 𝜖 𝔽, an additive map L : 𝒰 → 𝒰 is called an additive 𝜉-Lie derivation on 𝒰 if L([u, v]𝜉) = [L(u), v]𝜉 + [u,L(v)]𝜉, where [u, v]𝜉 = uv − 𝜉vu holds for all u, v 𝜖 𝒰. In the present paper, under certain assumptions imposed on 𝒰, it is shown that every Lie derivation (resp., additive 𝜉-Lie derivation) L on U is of standard form, i.e., L = δ+∅, where δ is an additive derivation on 𝒰 and ∅ is a mapping ∅ : 𝒰 → Z(𝒰) vanishing at [u, v] with uv = 0 in 𝒰. Moreover, we also characterize the additive 𝜉-Lie derivation for 𝜉 6= 1 by its action on zero product in a unital algebra over F.
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References
D. Benkovič and N. Širovnik, “Jordan derivations of unital algebras with idempotents,” Linear Algebra Appl., 437, 2271–2284 (2012).
D. Benkoviˇc, “Lie triple derivations of unital algebras with idempotents,” Linear Multilinear Algebra, 63, No. 1, 141–165 (2015).
J. A. Brooke, P. Busch, and D. B. Pearson, “Commutativity up to a factor of bounded operators in complex Hilbert spaces,” R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458(2017), 109–118 (2002).
W. S. Cheung, Maps on Triangular Algebras, Ph. D. Dissertation, Univ. Victoria (2000).
P. Ji and W. Qi, “Characterizations of Lie derivations of triangular algebras,” Linear Algebra Appl., 435, 1137–1146 (2011).
P. Ji, W. Qi, and X. Sun, “Characterizations of Lie derivations of factor von Neumann algebras,” Linear Multilinear Algebra, 61, No. 3, 417–428 (2013).
C. Kassel, “Quantum groups,” Graduate Texts in Mathematics, 155, Springer-Verlag, New York (1995).
F. Lu and W. Jing, “Characterizations of Lie derivations of B(X),” Linear Algebra Appl., 432, 89–99 (2010).
X. Qi and J. Hou, “Additive Lie (𝜉-Lie) derivations and generalized Lie (𝜉-Lie) derivations on nest algebras,” Linear Algebra Appl., 431, 843–854 (2009).
X. Qi, J. Cui, and J. Hou, “Characterizing additive 𝜉-Lie derivations of prime algebras by 𝜉-Lie zero products,” Linear Algebra Appl., 434, 669–682 (2011).
X. Qi, “Characterizing Lie derivations on triangular algebras by local actions,” Electron. J. Linear Algebra, 26, 816–835 (2013).
X. Qi and J. Hou, “Characterization of Lie derivations on von Neumann algebras,” Linear Algebra Appl., 438, 533–548 (2013).
X. Qi, “Characterization of (generalized) Lie derivations on J -subspace lattice algebras by local action,” Aequat. Math., 87, 53–69 (2014).
X. Qi, J. Ji, and J. Hou, “Characterization of additive maps 𝜉-Lie derivable at zero on von Neumann algebras,” Publ. Math. Debrecen, 86, No. 1-2, 99–117 (2015). P. Šemrl, “Additive derivations of some operator algebras,” Illinois J. Math., 35, 234–240 (1991).
W. Yang and J. Zhun, “Characterizations of additive (generalized) 𝜉-Lie (𝛼, β)-derivations on triangular algebras,” Linear Multilinear Algebra, 61, No. 6, 811–830 (2013).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 4, pp. 455–466, April, 2021. Ukrainian DOI: 10.37863/10.37863/umzh.v73i4.838.
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Ashraf, M., Jabeen, A. Characterizations of Additive 𝜉-Lie Derivations on Unital Algebras. Ukr Math J 73, 532–546 (2021). https://doi.org/10.1007/s11253-021-01941-y
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DOI: https://doi.org/10.1007/s11253-021-01941-y