×

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 \(n\times n\) complex matrices, is the Lie product \([A, B] =AB-BA\), \(A,B \in M_n\). 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 \(n\times n\) 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.

MSC:

17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
15A04 Linear transformations, semilinear transformations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dolinar, G.Maps onMnpreserving Lie products. (Preprint)
[2] Šemrl, P.Non-linear commutativity preserving maps. (to appear inActa Sci.)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.