# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\left[A,B\right]=AB-BA$, $A,B\in {M}_{n}$. 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 $n×n$ 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 17B40 Automorphisms, derivations and other operators on Lie algebras 15A04 Linear transformations, semilinear transformations (linear algebra)