Algebre, določene z ničelnim produktom

doktorska disertacija

Mateja Grašič (Author), Matej Brešar (Mentor)

Abstract

V doktorski disertaciji so obravnavane algebre, določene z ničelnim produktom. Ta pojem je nov. Zato bo ve%cji del disertacije namenjen ugotavljanju določenosti z ničelnim produktom standardnih primerov asociativnih, Liejevih in jordanskih algeber. V prvem delu se osredotočimo na asociativne algebre in pokažemo, da je vsaka matrična algebra nad algebro z enoto določena z ničelnim produktom. Nato sledi obravnava multiaditivnih preslikav, ki zadoščajo določenemu pogoju ohranjanja ničelnih produktov. Opisano je obnašanje teh preslikav na podkolobarju, generiranem z vsemi idempotenti danega kolobarja. Poseben primer tega rezultata je v pomoč pri dokazu, da je vsaka enotska algebra, ki je generirana s svojimi idempotenti, določena z ničelnim produktom. Prav tako je vsaka končno razsežna enostavna algebra, ki ni obseg, določena z ničelnim produktom. Drugi del je namenjen Liejevim algebram. Dokažemo, da je z ničelnim Liejevim produktom določena vsaka matrična algebra nad enotsko asociativno algebro B, določeno z ničelnim Liejevim produktom. Podan je primer matrične algebre, ki pove, da je res treba dodati določene predpostavke na algebro B. V nadaljevanju tega poglavja je dokazano še, da sta z ničelnim produktom določeni tudi Liejevi algebri poševno simetričnih matrik glede na transponiranje in simplektično involucijo. V tretjem so obravnavane najbolj znane jordanske algebre. Dokazano je, da so z ničelnim jordanskim produktom določene: algebra matrik nad poljubno enotsko algebro, algebra simetričnih matrik glede na transponiranje in simplektično involucijo, Albertova algebra ter jordanska algebra, določena z nedegenerirano simetrično bilinearno formo. Zadnji del je namenjen obravnavi določenih aditivnih preslikav na prakolobarjih.

Keywords

algebra;bilinearna preslikava;funkcijska identiteta;matrike;Albertova algebra;homomorfizem;idempotent;jordanska algebra;Liejeva algebra;linearna preslikava;multiaditivna preslikava;prakolobar;poševnosimetrična matrika;simetrična matrika;simlektična involucija;transponiranje;algebra, določena z ničelnim produktom;

Data

Language:	Slovenian
Year of publishing:	2012
Source:	Ljubljana
Typology:	2.08 - Doctoral Dissertation
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[M. Grašič]
UDC:	512.643(043.3)
COBISS:	260808704
Views:	3121
Downloads:	234
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Zero product determined algebras
Secondary abstract:	The central object in this thesis is a zero product determined algebra. This concept is new. The bulk of the thesis is therefore devoted to the question whether standard examples of associative, Lie, and Jordan algebras are zero product determined. In the first part we treat associative algebras. We show that the matrix algebra over any unital algebra is zero product determined. Next we deal with multiadditive maps satisfying a certain zero product preserver condition. Their form is described on the subring generated with all idempotents of the ring in question. This result implies that every unital algebra generated with idempotents is zero product determined. We also show that every simple finite dimensional algebra, which is not a division algebra, is zero product determined. The second chapter is devoted to Lie algebras. We show that if an associative algebra B is zero Lie product determined, then so is the matrix algebra over B. By an example we justify the assumption that B must be zero Lie product determined. We conclude this part by showing that Lie algebras of skew symmetric matrices with respect to either transpose or symplectic involution are zero product determined. Next we turn our attention to Jordan algebras. We prove that the algebra of matrices over a unital algebra endowed with the Jordan product, the Jordan algebra of symmetric matrices with respect to either transpose or symplectic involution, the Albert algebra, and the Jordan algebra determined with a nondegenerate symmetric bilinear form are all zero product determined. At the end we deal with additive maps between prime rings.
Secondary keywords:	algebra;bilinear map;functional identity;matrix;zero product;dissertations;Matrična algebra;Disertacije;Ničelni produkt;
URN:	URN:SI:UM:
Type (COBISS):	Dissertation
Thesis comment:	Univ. Maribor, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	95 str.
Keywords (UDC):	mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;
ID:	20462