1. Jordan maps and zero Lie product determined algebrasMatej Brešar, 2022, original scientific article Abstract: Let ▫$A$▫ be an algebra over a field ▫$F$▫ with ▫$\mathrm{char} (F) \ne 2$▫. If ▫$A$▫ is generated as an algebra by ▫$[[A,A],[A,A]]$▫, then for every skew-symmetric bilinear map ▫$\Phi:A \times A \to X$▫, where ▫$X$▫ is an arbitrary vector space over ▫$F$▫, the condition that ▫$\Phi(x^2,x)=0$▫ for all ▫$x \in A$▫ implies that ▫$\Phi(xy,z) +\Phi(zx,y) + \Phi(yz,x)=0$▫ for all ▫$x,y,z \in A$▫. This is applicable to the question of whether ▫$A$▫ is zero Lie product determined, and is also used in proving that a Jordan homomorphism from ▫$A$▫ onto a semiprime algebra ▫$B$▫ is the sum of a homomorphism and an antihomomorphism.
Keywords: bilinear map, zero Lie product determined algebra, derivation, Jordan derivation, Jordan homomorphism, functional identity
Published in DKUM: 18.08.2023; Views: 421; Downloads: 46
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5. On bilinear maps on matrices with applications to commutativity preserversMatej Brešar, Peter Šemrl, 2006, original scientific article Abstract: Let ▫$M_n$▫ be the algebra of all ▫$n times n$▫ matrices over a commutative unital ring ▫$mathcal{C}$▫, and let ▫$mathcal{L}$▫ be a ▫$mathcal{C}$▫-module. Various characterizations of bilinear maps ▫${,.,,,.,}: M_n times M_n to mathcal{L}$▫ with the property that ▫${x,y} = 0$▫ whenever ▫$x$▫ any ▫$y$▫ commute are given. As the main application of this result we obtain the definitive solution of the problem of describing (not necessarily bijective) commutativity preserving linear maps from ▫$M_n$▫ into ▫$M_n$▫ for the case where ▫$mathcal{C}$▫ is an arbitrary field; moreover, this description is valid in every finite dimensional central simple algebra.
Keywords: mathematics, matrix algebra, central simple algebra, functional identity, nonassociative product, Lie-admissible algebra, commutativity preserving map
Published in DKUM: 10.07.2015; Views: 1315; Downloads: 105
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6. Benkovic, Dominik; Eremita, Daniel: Commuting traces and commutativity preserving maps on triangular algebras. (English). - [J] J. Algebra 280, No.2, 797-824 (2004). [ISSN 0021-8693]Matej Brešar, 2006, review, book review, critique Keywords: matematika, algebra, trikotne algebre, gnezdne algebre, komutirajoče sledi, bilinearne preslikave, Liejevi izomorfizmi, ohranjevalci komutativnosti, mathematics, algebra, triangular algebras, nest algebras, commuting traces, bilinear maps, Lie isomorphisms, commutativity preserving maps
Published in DKUM: 10.07.2015; Views: 1174; Downloads: 35
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7. Commuting maps: a surveyMatej Brešar, 2004, review article Abstract: A map ▫$f$▫ on a ring ▫$mathcal{A}$▫ is said to be commuting if ▫$f(x)$▫ commutes with ▫$x$▫ for every ▫$x in mathcal{A}$▫. The paper surveys the development of the theory of commuting maps and their applications. The following topics are discussed: commuting derivations, commuting additive maps, commuting traces of multiadditive maps, various generalizations of the notion of a commuting map, and applications of results on commuting maps to different areas, in particular to Lie theory.
Keywords: matematika, algebra, prakolobar, komutirajoča preslikava, funkcijska identiteta, Banachova algebra, odvajanje, Liejeve algebre, linearni ohranjevalci, mathematics, algebra, commuting map, functional identity, prime ring, Banach algebra, derivation, Lie theory, linear preservers
Published in DKUM: 10.07.2015; Views: 1541; Downloads: 33
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