Automorphisms and derivations of finite-dimensional algebras
Brešar, Matej
Naj bo ▫$A$▫ končno-razsežna algebra nad poljem ▫$F$▫ z lastnostjo char▫$(F)\ne 2$▫. Če linearna preslikava ▫$D:A\to A$▫ zadošča ▫$xD(x)x\in [A,A]$▫ za vsak ▫$x\in A$▫, potem je ▫$D$▫ vsota ...notranjega odvajanja in linearne preslikave s sliko v radikalu ▫$A$▫. Če dodatno privzamemo, da je ▫$A$▫ polenostavna in char▫$(F)\ne 3$▫, potem linearna preslikava ▫$T:A\to A$▫ zadošča ▫$T(x)^3- x^3 \in [A,A]$▫ za vsak ▫$x\in A$▫ natanko tedaj, ko obstajata tak jordanski avtomorfizem ▫$J$▫ v multiplikacijski algebri ▫$A$▫ in tak centralen element ▫$\alpha$▫ z lastnostjo ▫$\alpha^3=1$▫, da je ▫$T(x)=\alpha J(x)$▫ za vse ▫$x\in A$▫. Ta rezultata sta uporabljena pri študiju lokalnih odvajanj in lokalnih (jordanskih) avtomorfizmov. Iz drugega rezultata tako izpeljemo, da je vsak lokalni jordanski avtomorfizem končno-razsežne enostavne algebre ▫$A$▫ (nad poljem s char▫$(F)\ne 2,3$▫) jordanski avtomorfizem.
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