## Lie-type derivations of finitary incidence algebras.(English)Zbl 1445.16039

The authors suppose that $$P$$ is an arbitrary partially ordered set, $$R$$ is a commutative ring with identity and $$\operatorname{FI}(P,R)$$ is the finitary incidence algebra of $$P$$ over $$R$$. Under some natural assumption on $$R$$ they prove that each Lie-type derivation of $$\operatorname{FI}(P,R)$$ is proper, which partially generalizes the main results of X. Zhang and the second author [Linear Algebra Appl. 513, 69–83 (2017; Zbl 1349.16074)], D. Wang and Z. Xiao [Commun. Algebra 47, No. 5, 1841–1852 (2019; Zbl 1456.16040)], and Z. Xiao and Y. Yang [Commun. Algebra 48, No. 1, 105–118 (2020; Zbl 1437.16041)]. In fact, the main goal of this paper is to describe Lie-type derivations of the finitary incidence algebra $$\operatorname{FI}(P,R)$$. Some of the results of this paper represent generalizations of other results. Also, they introduce some definitions which employ in their results. Taking into account the definitions of Lie derivations and Lie triple derivations, one naturally expects to extend them in one more general way via a Lie $$n$$-derivation. The authors employ Lie $$n$$-derivation on $$\operatorname{FI}(P,R)$$. The authors close their article by an open problem. They point out that all involved Lie-type derivations in this work are linear. They state the following theorem.
Theorem 2.13. Let $$L$$ be a Lie $$n$$-derivation of $$\operatorname{FI}(P,R)$$. If $$R$$ is $$(n-1)$$-torsion-free, then $$L=d+\tau$$, where $$d$$ is a derivation of $$\operatorname{FI}(P,R)$$ and $$\tau$$ is a central-valued linear mapping which annihilates all the $$n$$-th commutators.
It is natural to ask whether Theorem 2.13 without the assumption of additivity. That is, it is interesting to investigate multiplicative Lie-type derivations of finitary incidence algebras, which are motivated by the following papers: [D. Benkovič and D. Eremita, Linear Algebra Appl. 436, No. 11, 4223–4240 (2012; Zbl 1247.16040); A. Fošner et al., Colloq. Math. 132, No. 1, 53–71 (2013; Zbl 1306.47047); Y. Wang and Y. Wang, Linear Algebra Appl. 438, No. 5, 2599–2616 (2013; Zbl 1272.16039)].

### MSC:

 16W25 Derivations, actions of Lie algebras 16S60 Associative rings of functions, subdirect products, sheaves of rings 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 47L35 Nest algebras, CSL algebras
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### References:

 [1] I. Z. Abdullaev, “$$n$$-Lie derivations on von Neumann algebras”, Uzbek. Mat. Zh. 5-6 (1992), 3-9. \xoxMR1258290 [2] K. Baclawski, “Automorphisms and derivations of incidence algebras”, Proc. Amer. Math. Soc. 36 (1972), 351-356. \xoxMR313133 \xoxZBL0233.06001 · Zbl 0233.06001 [3] D. Benkovič, “Lie derivations on triangular matrices”, Linear Multilinear Algebra 55:6 (2007), 619-626. \xoxMR2361566 \xoxZBL1129.16024 · Zbl 1129.16024 [4] D. Benkovič, “Lie triple derivations on triangular matrices”, Algebra Colloq. 18:Special Issue 1 (2011), 819-826. \xoxMR2860364 \xoxZBL1297.16039 · Zbl 1297.16039 [5] D. Benkovič, “Lie triple derivations of unital algebras with idempotents”, Linear Multilinear Algebra 63:1 (2015), 141-165. \xoxMR3273744 \xoxZBL1315.16037 · Zbl 1315.16037 [6] D. Benkovič \andword D. Eremita, “Commuting traces and commutativity preserving maps on triangular algebras”, J. Algebra 280:2 (2004), 797-824. \xoxMR2090065 \xoxZBL1076.16032 · Zbl 1076.16032 [7] D. Benkovič \andword D. Eremita, “Multiplicative Lie $$n$$-derivations of triangular rings”, Linear Algebra Appl. 436:11 (2012), 4223-4240. \xoxMR2915278 \xoxZBL1247.16040 · Zbl 1247.16040 [8] M. Brešar, “Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings”, Trans. Amer. Math. Soc. 335:2 (1993), 525-546. \xoxMR1069746 \xoxZBL0791.16028 · Zbl 0791.16028 [9] M. Brešar, “Commuting maps: a survey”, Taiwanese J. Math. 8:3 (2004), 361-397. \xoxMR2163313 \xoxZBL1078.16032 · Zbl 1078.16032 [10] R. Brusamarello \andword D. W. Lewis, “Automorphisms and involutions on incidence algebras”, Linear Multilinear Algebra 59:11 (2011), 1247-1267. \xoxMR2847351 \xoxZBL1247.16038 · Zbl 1247.16038 [11] R. Brusamarello, E. Z. Fornaroli\serialcomma \andword E. A. Santulo Júnior, “Classification of involutions on finitary incidence algebras”, Internat. J. Algebra Comput. 24:8 (2014), 1085-1098. \xoxMR3296357 \xoxZBL1319.16027 · Zbl 1319.16027 [12] W.-S. Cheung, “Lie derivations of triangular algebras”, Linear Multilinear Algebra 51:3 (2003), 299-310. \xoxMR1995661 \xoxZBL1060.16033 · Zbl 1060.16033 [13] Y. Du \andword Y. Wang, “Lie derivations of generalized matrix algebras”, Linear Algebra Appl. 437:11 (2012), 2719-2726. \xoxMR2964719 \xoxZBL1266.16046 · Zbl 1266.16046 [14] A. Fošner, F. Wei\serialcomma \andword Z. Xiao, “Nonlinear Lie-type derivations of von Neumann algebras and related topics”, Colloq. Math. 132:1 (2013), 53-71. \xoxMR3106088 \xoxZBL1306.47047 · Zbl 1306.47047 [15] I. N. Herstein, “Lie and Jordan structures in simple, associative rings”, Bull. Amer. Math. Soc. 67 (1961), 517-531. \xoxMR139641 \xoxZBL0107.02704 · Zbl 0107.02704 [16] I. Kaygorodov, M. Khrypchenko\serialcomma \andword F. Wei, “Higher derivations of finitary incidence algebras”, Algebr. Represent. Theory 22:6 (2019), 1331-1341. \xoxMR4034785 \xoxZBL07152722 · Zbl 1456.16026 [17] N. Khripchenko, “Automorphisms of finitary incidence rings”, Algebra Discrete Math. 9:2 (2010), 78-97. \xoxMR2808782 \xoxZBL1224.18008 · Zbl 1224.18008 [18] N. S. Khripchenko, “Derivations of finitary incidence rings”, Comm. Algebra 40:7 (2012), 2503-2522. \xoxMR2948843 \xoxZBL1266.16030 · Zbl 1266.16030 [19] N. S. Khripchenko \andword B. V. Novikov, “Finitary incidence algebras”, Comm. Algebra 37:5 (2009), 1670-1676. \xoxMR2526329 \xoxZBL1180.16021 · Zbl 1180.16021 [20] M. Khrypchenko, “Jordan derivations of finitary incidence rings”, Linear Multilinear Algebra 64:10 (2016), 2104-2118. \xoxMR3521160 \xoxZBL1377.16037 · Zbl 1377.16037 [21] M. Khrypchenko, “Local derivations of finitary incidence algebras”, Acta Math. Hungar. 154:1 (2018), 48-55. \xoxMR3746522 \xoxZBL1399.16106 · Zbl 1399.16106 [22] M. Koppinen, “Automorphisms and higher derivations of incidence algebras”, J. Algebra 174:2 (1995), 698-723. \xoxMR1334232 \xoxZBL0835.16029 · Zbl 0835.16029 [23] F. Lu, “Lie derivations of certain CSL algebras”, Israel J. Math. 155 (2006), 149-156. \xoxMR2269428 \xoxZBL1130.47055 · Zbl 1130.47055 [24] F. Lu, “Lie derivations of $$\mathscr{I}$$-subspace lattice algebras”, Proc. Amer. Math. Soc. 135:8 (2007), 2581-2590. \xoxMR2302579 \xoxZBL1116.47060 · Zbl 1116.47060 [25] F. Lu \andword B. Liu, “Lie derivations of reflexive algebras”, Integral Equations Operator Theory 64:2 (2009), 261-271. \xoxMR2516282 \xoxZBL1201.47078 · Zbl 1201.47078 [26] W. S. Martindale, III, “Lie derivations of primitive rings”, Michigan Math. J. 11 (1964), 183-187. \xoxMR166234 \xoxZBL0123.03201 · Zbl 0123.03201 [27] M. Mathieu \andword A. R. Villena, “The structure of Lie derivations on $$C^\ast$$-algebras”, J. Funct. Anal. 202:2 (2003), 504-525. \xoxMR1990536 · Zbl 1032.46086 [28] C. R. Miers, “Lie derivations of von Neumann algebras”, Duke Math. J. 40 (1973), 403-409. \xoxMR315466 \xoxZBL0264.46064 · Zbl 0264.46064 [29] C. R. Miers, “Lie triple derivations of von Neumann algebras”, Proc. Amer. Math. Soc. 71:1 (1978), 57-61. \xoxMR487480 \xoxZBL0384.46047 · Zbl 0384.46047 [30] X. Qi, “Characterizing Lie $$n$$-derivations for reflexive algebras”, Linear Multilinear Algebra 63:8 (2015), 1693-1706. \xoxMR3305003 \xoxZBL1307.47037 · Zbl 1307.47037 [31] X. Qi, “Lie $$n$$-derivations on $$\mathcal{J}$$-subspace lattice algebras”, Proc. Indian Acad. Sci. Math. Sci. 127:3 (2017), 537-545. \xoxMR3660350 \xoxZBL06767432 · Zbl 06767432 [32] E. Spiegel, “On the automorphisms of incidence algebras”, J. Algebra 239:2 (2001), 615-623. \xoxMR1832909 \xoxZBL0996.16024 · Zbl 0996.16024 [33] E. Spiegel \andword C. J. O’Donnell, Incidence algebras, Monographs and Textbooks in Pure and Applied Mathematics 206, Marcel Dekker, New York, 1997. \xoxMR1445562 \xoxZBL0871.16001 [34] R. P. Stanley, “Structure of incidence algebras and their automorphism groups”, Bull. Amer. Math. Soc. 76 (1970), 1236-1239. \xoxMR263718 \xoxZBL0205.31702 · Zbl 0205.31702 [35] R. P. Stanley, Enumerative combinatorics, Vol. 1, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, 1997. \xoxMR1442260 \xoxZBL0889.05001 [36] Y. Wang, “Lie $$n$$-derivations of unital algebras with idempotents”, Linear Algebra Appl. 458 (2014), 512-525. \xoxMR3231830 \xoxZBL1303.16043 · Zbl 1303.16043 [37] Y. Wang \andword Y. Wang, “Multiplicative Lie $$n$$-derivations of generalized matrix algebras”, Linear Algebra Appl. 438:5 (2013), 2599-2616. \xoxMR3005317 \xoxZBL1272.16039 · Zbl 1272.16039 [38] D. Wang \andword Z. Xiao, “Lie triple derivations of incidence algebras”, Comm. Algebra 47:5 (2019), 1841-1852. \xoxMR3977704 \xoxZBL07089648 · Zbl 1456.16040 [39] M. Ward, “Arithmetic functions on rings”, Ann. of Math. $$(2) 38$$:3 (1937), 725-732. \xoxMR1503363 \xoxZBL0017.19404 · JFM 63.0085.04 [40] Z. Xiao, “Jordan derivations of incidence algebras”, Rocky Mountain J. Math. 45:4 (2015), 1357-1368. \xoxMR3418198 \xoxZBL1328.16022 · Zbl 1328.16022 [41] Z. Xiao \andword F. Wei, “Lie triple derivations of triangular algebras”, Linear Algebra Appl. 437:5 (2012), 1234-1249. \xoxMR2942345 \xoxZBL1253.16042 · Zbl 1253.16042 [42] Z. Xiao \andword Y. Yang, “Lie $$n$$-derivations of incidence algebras”, Comm. Algebra 48:1 (2020), 105-118. · Zbl 1437.16041 [43] X. Zhang \andword M. Khrypchenko, “Lie derivations of incidence algebras”, Linear Algebra Appl. 513 (2017), 69-83. \xoxMR3573789 \xoxZBL1349.16074 · Zbl 1349.16074
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.