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| Author(s): |
Daniel Eiti Nishida Kawai
Total Authors: 1
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| Document type: | Doctoral Thesis |
| Press: | São Paulo. |
| Institution: | Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI) |
| Defense date: | 2025-02-21 |
| Examining board members: |
Bruno Leonardo Macedo Ferreira;
Maria Elisabete Félix Barreiro;
Dylene Agda Souza de Barros;
Fernando Araujo Borges;
Tanise Carnieri Pierin
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| Advisor: | Henrique Guzzo Junior; Bruno Leonardo Macedo Ferreira |
| Abstract | |
In this thesis, we present results regarding functional equations and additivity of functions in various algebras, some not necessarily associative. We describe all additive functions f,g:\\mathbbightarrow\\mathbb satisfying the identity F(x)+M(x)G(x^)=0 for all x eq0, where \\mathbb is a field and m:\\mathbbightarrow\\mathbb is a given multiplicative function. After this, we describe all biadditive functions T:V\\times Vightarrow\\mathbb, where V is a vector space over a field \\mathbb, that are functionally homogenized by a multiplicative function M, meaning that T(ax,ay)=M(a)T(x,y) for all a\\in\\mathbb and x,y\\in V. The case where \\mathrm(\\mathbb) eq2 was solved in the article ``The equation F(x)+M(x)G(1/x) = 0 and homogeneous biadditive forms\'\', then we study the case where \\mathrm(\\mathbb)=2. Moreover, we describe all additive functions f,g:Dightarrow D satisfying the identity f(x)+x^ng(x^)=0 for all invertible x, where n is a nonnegative integer and D is an alternative division algebra. The case where D is associative was solved in the articles ``Certain functional identities on division rings\'\' and ``Certain functional identities on division rings of characteristic two\'\', so we study the case where D is not associative. Also, we study this functional equation where D is a split octonion algebra. Later, we study the Jordan derivations and Lie derivations in alternative division algebras of characteristic not 2 and in split octonion algebras over fields of characteristic not 2. If D is such an algebra, we show that every Jordan derivation satisfying a certain additional identity is a derivation and every Lie derivation has the form \\delta+\\tau, where \\delta is a derivation in D and \\tau:Dightarrow Z(D) is an additive function such that \\tau([x,y])=0 for all x,y\\in D. These results mirror the results from the articles ``Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings\'\' and ``Commuting traces of biadditive maps revisited\'\'. Finally, we study the additivity of some kinds of functions in some algebras. We study the additivity of functions f:Aightarrow B that satisfy the identity \\varphi(\\{a,b\\}_*+b^*a)=\\{\\varphi(a),\\varphi(b)\\}_* + \\varphi(b)^*\\varphi(a) or the identity \\varphi(\\{a,b\\}_*+a^*b)=\\{\\varphi(a),\\varphi(b)\\}_* + \\varphi(a)^*\\varphi(b) in associative algebras with involution that have an idempotent element and satisfy certain conditions, thus extending the results of the article ``Mappings preserving sum of products a\\diamond b+b^*a (resp., a^*\\diamond b+ab^*) on *-algebras\'\'. After this, we study the additivity of n-multiplicative isomorphisms, n-multiplicative derivations, elementary functions and Jordan elementary functions in Jordan algebras and some axial algebras, thus extending the results of the articles ``Additivity of Jordan maps on Jordan algebras\'\', ``An approach between the multiplicative and additive structure of a Jordan ring\'\', ``Additivity of Jordan derivations on Jordan algebras with idempotents\'\' and ``Multiplicative isomorphisms and derivations on axial algebras\'\'. (AU) | |
| FAPESP's process: | 22/14579-0 - Sufficient conditions for isomorphism between isotopes of nonassociative algebras |
| Grantee: | Daniel Eiti Nishida Kawai |
| Support Opportunities: | Scholarships in Brazil - Doctorate |