20.500.12556/DKUM-16235 JORDANSKA ODVAJANJA IN JORDANSKI IZOMORFIZMI NA TRIKOTNIH ALGEBRAH JORDAN DERIVATIONS AND JORDAN ISOMORPHISMS ON TRIANGULAR ALGEBRAS V diplomskem delu so na trikotnih algebrah obravnavana jordanska odvajanja in jordanski izomorfizmi. Trikotna algebra A je algebra, ki je izomorfna algebri oblike A M B kjer sta A in B enotski algebri in M enotski (A; B)-bimodul. Osnovna primera trikotnih algeber sta algebra zgornje trikotnih matrik T_n(C) in gnezdna algebra T(N). Linearni preslikavi d iz algebre A v A-bimodul M pravimo jordansko odvajanje, če velja d (xy + yx) = d(x)y + xd(y) + d(y)x + yd(x) za vse x; y iz A. Jordanski homomorfiem iz algebre A v algebro B je linearna preslikava ', za katero velja ' (xy + yx) = ' (x) ' (y) + ' (y) ' (x) za vse x; y iz A. Za vsako odvajanje velja, da je tudi jordansko odvajanje. Pogoji, kadar velja tudi obrat, so predstavljeni v poglavju o jordanskih odvajanjih na trikotnih algebrah. Pokazano je, da je vsako jordansko odvajanje iz trikotne algebre A = Tri(A;M;B) vase odvajanje. V zadnjem poglavju so podani pogoji, ki morajo veljati, da sta algebra zgornje trikotnih matrik T_n(C) in gnezdna algebra T(N) nerazcepni. Trikotna algebra A = Tri(A;M;B) je nerazcepna, če modula M ni mogoče zapisati kot direktno vsoto dveh netrivialnih podmodulov. Na koncu diplomskega dela je dokazano, da je ob ustreznih predpostavkah vsak jordanski izomorfiem iz trikotne algebre A v neko drugo algebro izomofizem ali antiizomorfizem. The graduation thesis considers Jordan derivations and Jordan isomorphisms on triangular algebras. An algebra A is called a triangular algebra if it is isomorphic to the algebra of the form A M B where A and B are unital algebras and M is a unital (A; B)-bimodule. Upper triangular matrix algebras T_n(C) and nest algebras T (N) are most common examples of triangular algebras. A linear map d mapping from an algebra A into an A-bimodule M is called a Jordan derivation if d (xy + yx) = d(x)y + xd(y) + d(y)x + yd(x) for every x in A. A Jordan homomorphism from an algebra A into an algebra B is a linear map ' satisfying ' (xy + yx) = ' (x) ' (y) + ' (y) ' (x) for all x; y in A: Every derivation is also a Jordan derivation. In chapter 5 we consider conditions under which the converse holds true as well. It is shown, that every Jordan derivation from a triangular algebra A = Tri(A;M; B) into itself is a derivation. In the last chapter it is shown which conditions needs to hold, that an upper triangular matrix algebra T_n(C) and a nest algebra T(N) are indecomposable. A triangular algebra A = Tri(A;M; B) is indecomposable if module M cannot be written as a direct sum of two nonzero submodules. At the end of the graduation thesis we show, that under certain assumptions every Jordan isomorphism from a triangular algebra A into some other algebra is either an isomorphism or an anti-isomorphism. trikotna algebra trikotna matrična algebra gnezdna algebra odvajanje jordansko odvajanje jordanski izomorfizem. triangular algebra triangular matrix algebra nest algebra derivation Jordan derivation Jordan isomorphism. true true false [I. Cizerl] Slovenski jezik Angleški jezik Diplomsko delo 2010-10-06 12:34:03 2010-11-10 16:21:30 2022-04-13 07:59:32 2023-12-20 03:24:07 2010 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 51(043.2) 17964040 URN:SI:UM:DK:6TVBXW6K UNI_Cizerl_Igor_2010.pdf UNI_Cizerl_Igor_2010.pdf 1 663E6A099B776158EC895DDA0F5939A2 232fe3795c73731004e9ce55e0f1bde9ee0bfd3faeece81c797ffce97d2f6cc7 998cde31-7c04-11eb-bb7a-00155d0001ca 20.500.12556/dkum/5168b428-2ae5-472a-bf17-b4316c6a9b11 https://dk.um.si/Dokument.php?lang=slv&id=18308 Fakulteta za naravoslovje in matematiko 0 0 0