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Commutative multiplication in three-dimensional space almost everywhere
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| Organization: | Belarussian State University |
| Department: | Chair of mathematical cybernetics |
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It is known that finite-dimensional division algebras over \[DoubleStruckCapitalR] exist only in dimensions 1, 2, 4, 8. In this work the nontrivial example of commutative group in {(x,y,z)\[Element][\[DoubleStruckCapitalR]^3:x^2+y^2>0} will be constructed. Multiplication operation in this group is not co-ordinated with usual addition operation in \[DoubleStruckCapitalR]^3.
Operations of multiplication and division of vectors X, Y I define following formulas: X={x1,x2,x3}, Y={y1,y2,y3},
X⊗Y={(Sqrt[x1^2+x2^2+x3^2] (x1 y1-x2 y2) Sqrt[y1^2+y2^2+y3^2])/(x3 y3+Sqrt[x1^2+x2^2+x3^2] Sqrt[y1^2+y2^2+y3^2]),(Sqrt[x1^2+x2^2+x3^2] (x2 y1+x1 y2) Sqrt[y1^2+y2^2+y3^2])/(x3 y3+Sqrt[x1^2+x2^2+x3^2] Sqrt[y1^2+y2^2+y3^2]),(Sqrt[x1^2+x2^2+x3^2] Sqrt[y1^2+y2^2+y3^2] (Sqrt[x1^2+x2^2+x3^2] y3+x3 Sqrt[y1^2+y2^2+y3^2]))/(x3 y3+Sqrt[x1^2+x2^2+x3^2] Sqrt[y1^2+y2^2+y3^2])},
X/Y={(Sqrt[x1^2+x2^2+x3^2] (x1 y1+x2 y2))/(Sqrt[y1^2+y2^2+y3^2] (-x3 y3+Sqrt[x1^2+x2^2+x3^2] Sqrt[y1^2+y2^2+y3^2])),(Sqrt[x1^2+x2^2+x3^2] (x2 y1-x1 y2))/(Sqrt[y1^2+y2^2+y3^2] (-x3 y3+Sqrt[x1^2+x2^2+x3^2] Sqrt[y1^2+y2^2+y3^2])),(Sqrt[x1^2+x2^2+x3^2] (-Sqrt[x1^2+x2^2+x3^2] y3+x3 Sqrt[y1^2+y2^2+y3^2]))/(Sqrt[y1^2+y2^2+y3^2] (-x3 y3+Sqrt[x1^2+x2^2+x3^2] Sqrt[y1^2+y2^2+y3^2]))}.
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Abelian group, three-dimensional arithmetical space
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Russian
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