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On the normalizer of the reflexive cover of a unital algebra of linear transformations
Bračič, Janko ; Kandić, Marko
Za dano unitalno algebro ▫${\mathcal A}$▫ linearnih preslikav na končno dimenzionalnem vektorskem prostoru ▫$V$▫ študiramo v tem članku množico ▫$\mathrm{Col}({\mathcal A})$▫ vseh tistih obrnljivih ...linearnih preslikav ▫$S$▫ na ▫$V$▫, ki preslikajo vsak podprostor ▫$M\in Lat({\mathcal A})$▫ v podprostor ▫$SM\in \mathrm{Lat}({\mathcal A})$▫. Pokažemo, da je ▫$Col({\mathcal A})$▫ normalizator grupe obrnljivih linearnih preslikav v refleksivnem pokritju ▫${\mathcal A}$▫. Za unitalno algebro ▫$(A)$▫, ki je generirana z linearno preslikavo ▫$A$▫, podamo popolni opis ▫$\mathrm{Col}(A)$▫. Najprej z uporabo primarne dekompozicije preslikave ▫$A$▫ reduciramo problem karakterizacije ▫$\mathrm{Col}(A)$▫ na problem karakterizacije ▫$\mathrm{Col}(N)$▫ dane nilpotentne linearne preslikave ▫$N$▫. Grupa ▫$\mathrm{Col}(N)$▫ vsebuje vse obrnljive preslikave iz komutanta ▫$(N)'$▫ in je vsebovana v grupi obrnljivih preslikav refleksivnega pokritja komutanta ▫$(N)'$▫. Dokažemo, da je ▫$\mathrm{Col}(N)$▫ prava podgrupa v ▫$\mathrm{(AlgLat}(N)')^{-1}$▫ natanko tedaj ko sta vsaj dve jordanski kletki v jordanski dekompoziciji ▫$N$▫ dimenzije ▫$2$▫ ali več. Na koncu izračunamo grupo ▫$\mathrm{Col}(J_2 \oplus J_2)$▫.
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