Frobenius theorem (real division algebras)

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Frobenius theorem (real division algebras) From: STALKER2002 Subject: General Date/Time 2008-07-08 17:13:29 Remote IP: 79.139.140.102 Message From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite dimensional associative division algebras over the real numbers. The theorem proves that the only associative division algebra which is not commutative over the real numbers is the quaternions. If D is a finite dimensional division algebra over the real numbers R then one of the following cases holds D = R D = C ( complex numbers) D isomorphic to H (quaternions) [edit] Pontryagin variant If D is a connected, locally compact division ring, then either D=R, or D=C, or D=H. [edit] References Ferdinand Georg Frobenius (1878) "Ueber lineare Substitutionen und bilineare Formen", Journal für die reine und angewandte Mathematik 84:1-63 (Crelle's Journal). Reprinted in Gesammelte Abhandlungen Band I, pp.343-405. Yuri Bahturin (1993) Basic Structures of Modern Algebra, Kluwer Acad. Pub. pp.30-2 [ISBN 0-7923-2459-5]. R.S. Palais (1968) "The Classification of Real Division Algebras" American Mathematical Monthly 75:366-8. Lev Semenovich Pontryagin, Topological Groups, page 159, 1966. Retrieved from "http://en.wikipedia.org/wiki/Frobenius_theorem_%28real_division_algebras%29" Categories: Abstract algebra | Quaternions | Mathematical theorems http://en.wikipedia.org/wiki/Frobenius_theorem_%28real_division_algebras%29 Geometrical Interpretation of Quaternions&MULTIPLICATION: (364) STALKER2002 (346) - - 2008-07-01 4:04 pm Re: Geometrical Interpretation of Complex Numbers&MULTIPLICATION: (276) STALKER2002 (346) - - 2008-07-07 00:06 am Multidimensional Generalization of Complex Numbers&Quaternions: (257) STALKER2002 (346) - - 2008-07-07 01:08 am Does anybody understand that this is a new DISCOVERY?: (313) STALKER2002 (346) - - 2008-07-07 2:59 pm NEW?: (272) Steven (680) - (Training Log) - 2008-07-08 4:20 pm Re: NEW? after quaternions and octanions there was NO other generalization of associative algebra: (261) STALKER2002 (346) - - 2008-07-08 4:39 pm Frobenius theorem (real division algebras): (279) STALKER2002 (346) - - 2008-07-08 5:13 pm Frobenius theorem (real division algebras): (255) Steven (680) - (Training Log) - 2008-07-08 5:18 pm Commutativity is not needed for quaternions and their generalization: (261) STALKER2002 (346) - - 2008-07-08 5:33 pm I never said that: (255) Steven (680) - (Training Log) - 2008-07-08 9:17 pm Not true: (263) Steven (680) - (Training Log) - 2008-07-08 5:14 pm The path AC is not the same as the pair of two arcs AB and BC: (242) STALKER2002 (346) - - 2008-07-08 5:22 pm Then how is your multiplication defined?: (248) Steven (680) - (Training Log) - 2008-07-08 9:31 pm Re: Then how is your multiplication defined? Look how!: (253) STALKER2002 (346) - - 2008-07-09 01:08 am Re: Then how is your multiplication defined? Look how!: (250) STALKER2002 (346) - - 2008-07-09 01:09 am Any composition of two rotations in R^4 is also a rotation in R^4 ,group SO(4)!: (255) STALKER2002 (346) - - 2008-07-09 01:28 am Any composition of two rotations in R^4 is also a rotation in R^4 ,group SO(4)!: (259) Steven (680) - (Training Log) - 2008-07-09 3:23 pm Not at All! The group of arcs deffers slightly from SO(4): (253) STALKER2002 (346) - - 2008-07-09 4:01 pm Not at All! The group of arcs differs slightly from SO(4): (264) STALKER2002 (346) - - 2008-07-09 4:05 pm However, some questions are still to be solved..: (261) STALKER2002 (346) - - 2008-07-09 5:00 pm Even in the AB@BC case, you have a well-defined problem (nt): (270) Steven (680) - (Training Log) - 2008-07-09 5:18 pm I know that; I wasn't talking about that (nt): (263) Steven (680) - (Training Log) - 2008-07-09 5:17 pm This is a problem: (271) Steven (680) - (Training Log) - 2008-07-09 3:13 pm The shortest way to Tokyo from Moscow is through China: (268) STALKER2002 (346) - - 2008-07-09 3:49 pm No, you can't force the continuity: (271) Steven (680) - (Training Log) - 2008-07-09 5:32 pm The second variation of functional length is zero on geodesic path: (274) STALKER2002 (346) - - 2008-07-10 01:56 am The FIRST, not the second variation of functional length is zero on geodesic path: (263) STALKER2002 (346) - - 2008-07-10 06:36 am That's my point!: (277) Steven (680) - (Training Log) - 2008-07-10 2:52 pm Of course Imust be more accurate and I care about geodesic preservation: (251) STALKER2002 (346) - - 2008-07-10 3:45 pm For any 3 points A, O, C there exists one and only one plane AOC,containing them,: (269) STALKER2002 (346) - - 2008-07-10 4:12 pm Yes: (256) Steven (680) - (Training Log) - 2008-07-10 11:15 pm Correct Idea! Because in 3D case I have defined multiplication of arcs via parallel translation of o: (266) STALKER2002 (346) - - 2008-07-11 00:17 am Antisymmetric 4-DYADS ARC[i,j]=N1(i)N2(j)-N2(i)N1(j). A New idea: (249) STALKER2002 (346) - - 2008-07-15 4:00 pm For simple 4-DYADS ARC1[i,j]=N1(i)N2(j) and ARC2[j,k]=N2(j)N3(k) ARC1@ARC2=N1(i)N3(k) : (234) STALKER2002 (346) - - 2008-07-15 4:51 pm This is why vector and tensor math is superior to and replaced quaternion interpretations (nt): (251) Steven (680) - (Training Log) - 2008-07-16 01:14 am By the WAY, in 2D case both of these arcs AC correspond to the same complex number!: (264) STALKER2002 (346) - - 2008-07-10 5:07 pm By the WAY, in 2D case both of these arcs AC correspond to the same complex number!: (260) Steven (680) - (Training Log) - 2008-07-10 10:49 pm [Top of List] [Previous Thread] [Next Thread] This Forum is for posting on topics of interest to the Taoist (Daoist) community. Tao is exceptionally broad. 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