U Vikipediyi ye statti pro inshi znachennya cogo termina Teorema Gurvica Teorema Gurvica pro kompozitni algebri teorema sho opisuye osnovni normovani algebri ne plutati z normovanimi banahovimi algebrami sho v funkcionalnomu analizi Cya teorema sformulovana nimeckim matematikom Adolfom Gurvicem v 1898 roci 1 Zmist 1 Viznachennya normovanoyi algebri 2 Formulyuvannya teoremi 3 Dovedennya 3 1 Lema 1 3 2 Lema 2 3 3 Dovedennya teoremi 4 Primitki 5 DzherelaViznachennya normovanoyi algebri RedaguvatiAlgebra nazivayetsya normovanoyu yaksho v nij mozhna vvesti skalyarnij dobutok z vlastivistyu a b a b a a b b displaystyle ab ab a a b b nbsp Oskilki vvivshi normu a a a 1 2 displaystyle a a a 1 2 nbsp otrimayemo a b a b displaystyle ab a cdot b nbsp Formulyuvannya teoremi RedaguvatiDovilna normovana algebra z odiniceyu izomorfna odnij z chotiroh algebr dijsnih chisel kompleksnih chisel kvaternioniv chi oktav Dovilna normovana algebra maye vlastivist alternativnosti a b b a b b b b a b b a displaystyle ab b a bb quad b ba bb a nbsp Dovedennya RedaguvatiLema 1 Redaguvati V dovilnij normovanij algebri spravedliva totozhnist a 1 b 1 a 2 b 2 a 1 b 2 a 2 b 1 2 a 1 a 2 b 1 b 2 displaystyle a 1 b 1 a 2 b 2 a 1 b 2 a 2 b 1 2 a 1 a 2 b 1 b 2 nbsp Lema 2 Redaguvati V dovilnij normovanij algebri z odiniceyu spravedliva totozhnist a b b b b a displaystyle ab bar b b b a nbsp Naslidkom lemi ye formula a x y a y x 2 x y a displaystyle ax bar y ay bar x 2 xy a nbsp Dovedennya teoremi Redaguvati Poznachimo odinicyu algebri A displaystyle mathcal A nbsp cherez 1 displaystyle mathbf 1 nbsp Kozhen element u A displaystyle u in mathcal A nbsp mozhna predstaviti yedinim chinom u viglyadi u k 1 u displaystyle u k mathbf 1 u prime nbsp de u 1 displaystyle u prime perp mathbf 1 nbsp Vvedemo v algebri operaciyu spryazhennya takim chinom u k 1 u displaystyle bar u k mathbf 1 u prime nbsp Nehaj U displaystyle mathcal U nbsp deyaka pidalgebra sho mistit 1 displaystyle mathbf 1 nbsp i ne zbigayetsya z A displaystyle mathcal A nbsp Todi isnuye odinichnij vektor e displaystyle e nbsp sho ortogonalnij do U displaystyle mathcal U nbsp Pokazhemo sho elementi vidu u 1 u 2 e u 1 u 2 U displaystyle u 1 u 2 e quad u 1 u 2 in mathcal U qquad qquad nbsp takozh utvoryuyut pidalgebru v A displaystyle mathcal A nbsp Poznachimo yiyi U U e displaystyle mathcal U U e nbsp Dlya cogo dovedemo Predstavlennya dovilnogo elementa z U U e displaystyle mathcal U U e nbsp u viglyadi mozhlive yedinim chinom Dovedennya vikoristovuye Lemu 1 Mnozhennya elementiv vidu zadovilnyaye formulu u 1 u 2 e v 1 v 2 e u 1 v 1 v 2 u 2 v 2 u 1 u 2 v 1 e displaystyle u 1 u 2 e v 1 v 2 e u 1 v 1 bar v 2 u 2 v 2 u 1 u 2 bar v 1 e nbsp yaka zbigayetsya z proceduroyu podvoyennya Keli Diksona Spochatku za dopomogoyu naslidku Lemi 2 dovedemo formuli u e v u v e u v e v u e u e v e v u displaystyle ue v u bar v e u ve vu e ue ve bar v u nbsp Z yakih legko otrimati danu formulu Dovilna pidalgebra U displaystyle mathcal U nbsp sho mistit 1 displaystyle mathbf 1 nbsp i ne zbigayetsya z A displaystyle mathcal A nbsp ye asociativnoyu Dovedennya vikoristovuye naslidok Lemi 2 Otzhe oskilki algebra A displaystyle mathcal A nbsp mistit odinicyu to v neyi ye pidalgebra z elementiv vidu k 1 displaystyle k mathbf 1 nbsp sho izomorfna algebri dijsnih chisel R displaystyle mathbb R nbsp Yaksho R displaystyle mathbb R nbsp ne zbigayetsya z algebroyu A displaystyle mathcal A nbsp to rozglyanemo pidalgebru C R R e displaystyle mathbb C mathbb R mathbb R e nbsp sho izomorfna algebri kompleksnih chisel Yaksho C displaystyle mathbb C nbsp ne zbigayetsya z algebroyu A displaystyle mathcal A nbsp to rozglyanemo pidalgebru Q C C e displaystyle mathbb Q mathbb C mathbb C e prime nbsp sho izomorfna algebri kvaternioniv Yaksho Q displaystyle mathbb Q nbsp ne zbigayetsya z algebroyu A displaystyle mathcal A nbsp to rozglyanemo pidalgebru O Q Q e displaystyle mathbb O mathbb Q mathbb Q e prime prime nbsp sho izomorfna algebri oktav Algebra O displaystyle mathbb O nbsp vzhe povinna zbigatisya z algebroyu A displaystyle mathcal A nbsp oskilki vona vzhe ne ye asociativnoyu Primitki Redaguvati Hurwitz A 1898 Uber die Composition der quadratischen Formen von beliebig vielen Variabeln Goett Nachr 309 316 Dzherela RedaguvatiKantor I L Solodovnikov A S Giperkompleksnye chisla Moskva Nauka 1973 144 s ros Otrimano z https uk wikipedia org w index php title Teorema Gurvica pro kompozitni algebri amp oldid 36798400
