Biderivations of ${\mathfrak s}{\mathfrak l}(2)$ on All Simple Modules in Prime Characteristic

Abstract

Over an algebraically closed field of characteristic p > 2, this paper gives a sufficient and necessary condition for a map from the Cartesian product of a finite-dimensional Lie algebra with itself two times to its any nontrivial and simple module to be a symmetric biderivation, and determines all biderivations of the 3-dimensional simple Lie algebra ${\mathfrak s}{\mathfrak l}(2)$ on its any finite-dimensional simple module.

Keywords

Biderivations / ${\mathfrak s}{\mathfrak l}(2)$')">${\mathfrak s}{\mathfrak l}(2)$ / simple modules / 17B05 / 17B10 / 17B40 / 17B50

References

[1] BrešarM. On generalized biderivations and related maps. J. Algebra, 1995, 172(3): 764-786.

[2] BrešarM, ZhaoK. Biderivations and commuting linear maps on Lie algebras. J. Lie Theory, 2018, 28(3): 885-900

[3] ChenZ. Biderivations and linear commuting maps on simple generalized Witt algebras over a field. Electron. J. Linear Algebra, 2016, 31: 1-12.

[4] ChiL, SunJ, YangH. Biderivations and linear commuting maps on the planar Galilean conformal algebra. Linear Multilinear Algebra, 2018, 66(8): 1606-1618.

[5] HanX, WangD, XiaC. Linear commuting maps and biderivations on the Lie algebras W(a,b). J. Lie Theory, 2016, 26(3): 777-786

[6] StradeH, FarnsteinerRModular Lie Algebras and Their Representations, 1988, New York. Marcel Dekker, Inc.. 116

[7] TangX. Biderivations, linear commuting maps and commutative post-Lie algebra structures on W-algebras. Comm. Algebra, 2017, 45(12): 5252-5261.

[8] TangX. Biderivations of finite-dimensional complex simple Lie algebras. Linear Multilinear Algebra, 2018, 66(2): 250-259.

[9] WangD, YuX. Biderivations and linear commuting maps on the Schrödinger–Virasoro Lie algebra. Comm. Algebra, 2013, 41(6): 2166-2173.

[10] WangD, YuX, ChenZ. Biderivations of the parabolic subalgebras of simple Lie algebras. Comm. Algebra, 2011, 39(11): 4097-4104.

[11] WangS, LiZ. Cohomology of ${\mathfrak s}{\mathfrak l}(2)$ and its applications in prime characteristic. Front. Math., 2024, 19(2): 271-282.

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Peking University