Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras

Marcoux, Laurent W.; Sourour, A. R.

doi:10.1016/s0024-3795(98)10182-9

Cited by 43 publications

(23 citation statements)

References 6 publications

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“…Here c is a nonzero element in F , T an invertible matrix, and f a linear function on M n (F ). In 1999, Marcoux et al [6] described commutativity preserving maps on T n (F ) of all upper triangular matrices, and in 2002, Cao et al [2] determined commutativity preserving maps on N n (F ) of all strictly upper triangular matrices. It was Omladič who first considered commutativity preserving maps on infinite-dimensional algebras.…”

Section: Introductionmentioning

confidence: 99%

Invertible linear maps on simple Lie algebras preserving commutativity

Wang

Chen

2011

Proc. Amer. Math. Soc.

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Abstract. Let g be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field of characteristic zero. An invertible linear map ϕ on g is called preserving commutativity in both directions if, for any x, y ∈ g,The group of all such maps on g is denoted by P zp(g). It is shown in this paper that, if l = 1, then P zp(g) = GL(g); otherwise, P zp(g) = Aut(g) × F * I g , where F * I g denotes the group of all nonzero scalar multiplication maps on g.

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Section: Introductionmentioning

confidence: 99%

Invertible linear maps on simple Lie algebras preserving commutativity

Wang

Chen

2011

Proc. Amer. Math. Soc.

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“…When we investigate the above-mentioned mappings, the principal task is to describe their forms. This is demonstrated by various works, see [8,10,11,12,13,15,20,21,28,29,33,36,41,44,45,49,50]. We encourage the reader to read the well-written survey paper [13], in which the author presented the development of the theory of semi-centralizing mappings and their applications in details.…”

Section: Introductionmentioning

confidence: 84%

“…Doković [23] showed that every Lie automorphism of upper triangular matrix algebras T n (R) over a commutative ring R without nontrivial idempotents has the standard form as well. Marcoux and Sourour [33] classified the linear mappings preserving commutativity in both directions (i.e., [x, y] = 0 if and only if [f(x), f(y)] = 0) on upper triangular matrix algebras T n (F) over a field F. Such a mapping is either the sum of an algebra automorphism of T n (F) (which is inner) and a mapping into the center FI, or the sum of the negative of an algebra anti-automorphism and a mapping into the center FI. The classification of the Lie automorphisms of T n (F) is obtained as a consequence.…”

Section: Introductionmentioning

confidence: 99%

Centralizing traces and Lie triple isomorphisms on triangular algebras

Xiao

Wei

Fošner

2014

Linear and Multilinear Algebra

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Abstract. Let T be a triangular algebra over a commutative ring R and Z(T ) be the center of T . Suppose that q : T × T −→ T is an R-bilinear mapping and that Tq : : T −→ T is a trace of q. We describe the form of Tq satisfying the condition [Tq(T ), T ] ∈ Z(T ) for all T ∈ T . The question of when Tq has the proper form will be addressed. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism on T to be almost standard. As applications we characterize Lie triple isomorphisms of triangular matrix algebras and nest algebras. Some further research topics related to current work are proposed at the end of this article.

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“…Doković [33] showed that every Lie automorphism of upper triangular matrix algebras T n (R) over a commutative ring R without nontrivial idempotents has the standard form as well. Marcoux and Sourour [58] classified the linear mappings preserving commutativity in both directions (i.e., [x, y] = 0 if and only if [f(x), f(y)] = 0) on upper triangular matrix algebras T n (F) over a field F. Such a mapping is either the sum of an algebraic automorphism of T n (F) (which is inner) and a mapping into the center FI, or the sum of the negative of an algebraic anti-automorphism and a mapping into the center FI. The classification of the Lie automorphisms of T n (F) is obtained as a consequence.…”

Section: Introductionmentioning

confidence: 99%

Centralizing traces and Lie triple isomorphisms on generalized matrix algebras

Liang

Wei

Xiao

et al. 2014

Linear and Multilinear Algebra

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Abstract. Let G be a generalized matrix algebra over a commutative ring R and Z(G) be the center of G. Suppose that q : G × G −→ G is an R-bilinear mapping and Tq : G −→ G is the trace of q. We describe the form of Tq satisfying the condition [Tq(G), G] ∈ Z(G) for all G ∈ G. The question of when Tq has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism of G to be almost standard. As applications we characterize Lie triple isomorphisms of full matrix algebras, of triangular algebras and of certain unital algebras with nontrivial idempotents. Some topics for future research closely related to our current work are proposed at the end of this article.

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Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras

Cited by 43 publications

References 6 publications

Invertible linear maps on simple Lie algebras preserving commutativity

Invertible linear maps on simple Lie algebras preserving commutativity

Centralizing traces and Lie triple isomorphisms on triangular algebras

Centralizing traces and Lie triple isomorphisms on generalized matrix algebras

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