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Maps on upper triangular matrices preserving Lie products. (English) Zbl 1160.17014
One of the standard products which induces a structure of an algebra on M n , the space of all n×n complex matrices, is the Lie product [A,B]=AB-BA, A,BM n . Every Lie automorphism of M n has a nice form. Recently, P. Šemrl [Acta Sci. Math. 71, No. 3–4, 781–819 (2005; Zbl 1111.15002)] and the author [Publ. Math. 71, No. 3–4, 467–477 (2007; Zbl 1164.17015)] obtained a characterization also for non-linear Lie homomorphisms. Let F be an arbitrary field with characteristic zero, let T n be the Lie algebra of all n×n upper triangular matrices over F. It is the aim of this study to characterize maps which preserve the Lie product on T n . We can construct various maps on higher-dimensional algebras of upper triangular matrices which preserve the Lie product. If we assume that the map on T n which preserves Lie product is bijective, then we can obtain a nice characterization.

17B60Lie (super)algebras associated with other structures
17B40Automorphisms, derivations and other operators on Lie algebras
15A04Linear transformations, semilinear transformations (linear algebra)