diplomsko delo
Povzetek
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)- bi modul. 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 -bi modul 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 pod modulov. 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.
Ključne besede
matematika;jordanska odvajanja;izomorfizmi;trikotne algebre;gnezdna algebra;matrična algebra;diplomska dela;
Podatki
Jezik: |
Slovenski jezik |
Leto izida: |
2010 |
Izvor: |
Maribor |
Tipologija: |
2.11 - Diplomsko delo |
Organizacija: |
UM FNM - Fakulteta za naravoslovje in matematiko |
Založnik: |
[I. Cizerl] |
UDK: |
51(043.2) |
COBISS: |
17964040
|
Št. ogledov: |
2264 |
Št. prenosov: |
99 |
Ocena: |
0 (0 glasov) |
Metapodatki: |
|
Ostali podatki
Sekundarni jezik: |
Angleški jezik |
Sekundarni naslov: |
JORDAN DERIVATIONS AND JORDAN ISOMORPHISMS ON TRIANGULAR ALGEBRAS |
Sekundarni povzetek: |
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. |
Sekundarne ključne besede: |
triangular algebra;triangular matrix algebra;nest algebra;derivation;Jordan derivation;Jordan isomorphism.; |
URN: |
URN:SI:UM: |
Vrsta dela (COBISS): |
Diplomsko delo |
Komentar na gradivo: |
Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo |
Strani: |
IX, 40 f. |
Ključne besede (UDK): |
mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika; |
ID: |
8761875 |