Jordanska odvajanja in jordanski izomorfizmi na trikotnih algebrah

diplomsko delo

Igor Cizerl (Author), Dominik Benkovič (Mentor)

Abstract

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.

Keywords

matematika;jordanska odvajanja;izomorfizmi;trikotne algebre;gnezdna algebra;matrična algebra;diplomska dela;

Data

Language:	Slovenian
Year of publishing:	2010
Source:	Maribor
Typology:	2.11 - Undergraduate Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[I. Cizerl]
UDC:	51(043.2)
COBISS:	17964040
Views:	2264
Downloads:	99
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	JORDAN DERIVATIONS AND JORDAN ISOMORPHISMS ON TRIANGULAR ALGEBRAS
Secondary abstract:	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.
Secondary keywords:	triangular algebra;triangular matrix algebra;nest algebra;derivation;Jordan derivation;Jordan isomorphism.;
URN:	URN:SI:UM:
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	IX, 40 f.
Keywords (UDC):	mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;
ID:	8761875