On the multiplicative group of simple algebras and orthogonal groups of three dimensions

Akira HATTORI

doi:10.2969/jmsj/00430205

Akira HATTORI

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JOURNAL FREE ACCESS

1952 Volume 4 Issue 3-4 Pages 205-217

DOI https://doi.org/10.2969/jmsj/00430205

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Published: 1952 Received: March 24, 1952 Available on J-STAGE: August 29, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: August 29, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) Cf. M. Abe, Projective transformation groups over non-commutative fields, Sijo-Sugaku-Danwakai. 240 (1942) (in Japanese). J. Dieudonné, Les déterminants sur un corps non-commutatif, Bull. Soc. Math. Fr. vol. 71 (1943).
2) "In a division algebra A, the only subalgebras invariant under every inner auto-morphism are the whole A and the center of A." This theorem was extended to any sfield and was simply proved by R. Brauer and L. K. Hua. See, for example : R. Brauer, On a theorem of H. Cartan, Bull. Amer. Math. Soc. vol. 55 (1949), pp. 619-620.
3) A. Hattori, On invariant subrings, Japanese Journ. of Math. vol. 21 (1951), pp. 121-129. This paper is referred to as IS in this paper.
4) B. L, van der Waerden, Gruppen von linearen Transformationen, Ergebnisse, (1935), p. 27.
5) J. Dieudonné, Sur les groupes classiques, Act. Sci. et Ind., 1040, (1948), n°15 and n°16.
6) In the whole of this paper, we shall assume that the identity of every simple subalgebra considered is the same as the identity of A.
7) See Abe, loc, cit.; cf. also the proof of Theorem 1 of IS.
8) Cf. Artin, Nesbitt and Thrall, Rings with minimum condition, (1944), p. 66.
9) This fact has been proved in another way. See van der Waerden, loc. cit.
10) A theorem due to H. Ha. See: E. Witt, Theorie der quadratischen Formen in beliebigen Körpern, Crelle's Journal, vol. 176 (1937), pp. 31-44. Satz 19.
11) As is well known, this follows from the sum-relation between _??_-invariants of A.
12) Cf. T. Nakayama and Y. Matsushima, Über die multiplikative Gruppe einer _??_-adischen Divisionsalgebra, Proc. Imp. Acad. Jap. vol. 19 (1944), pp. 622-628.
13) A subspace E is called non.isotropic if the restriction of f on E is non-degenerate, and a vector v is called non-isotropic if the straight line {v} is non.isotropic. See, for example, J. Dieudonné, loc. cit.
14) If b is non-isotropic, take a non-isotropic c from the orthogonal complement of {b}, if b is isotropic, take a vector c not orthogonal to b, then U={b, c} is the desired one.
15) After leading the first manuscript of this paper which had been consisting of §§ 1 and 2 of the present paper, Mr. N. Iwahori informed to the author the following result: Let A be an algebra over the real number field with the center Z, and assume that the group of inner automorphisms of A gives rise to the whole rotation group of the real vector space A/Z, then A/Z has the dimension three. Our Proposition 7 was obtained in generalizing this result. Mr. N. Iwabori found a proof of this proposition, in case of characteristic zero, by a completely different argument.

Right : 1) Cf. M. Abe, Projective transformation groups over non-commutative fields, Sijo-Sugaku-Danwakai. 240 (1942) (in Japanese). J. Dieudonné, Les déterminants sur un corps non-commutatif, Bull. Soc. Math. Fr. vol. 71 (1943).
2) “In a division algebra A, the only subalgebras invariant under every inner automorphism are the whole A and the center of A.” This theorem was extended to any sfield and was simply proved by R. Brauer and L. K. Hua. See, for example: R. Brauer, On a theorem of H. Cartan, Bull. Amer. Math. Soc. vol. 55 (1949), pp. 619-620.
3) A. Hattori, On invariant subrings, Japanese Journ. of Math. vol. 21 (1951), pp. 121-129. This paper is referred to as IS in this paper.
4) B. L. van der Waerden, Gruppen von linearen Transformationen, Ergebnisse, (1935), p. 27.
5) J. Dieudonné, Sur les groupes classiques, Act. Sci. et Ind., 1040, (1948), n°15 and n°16.
6) In the whole of this paper, we shall assume that the identity of every simple subalgebra considered is the same as the identity of A.
7) See Abe, loc. cit.; cf. also the proof of Theorem 1 of IS.
8) Cf. Artin, Nesbitt and Thrall, Rings with minimum condition, (1944), p. 66.
9) This fact has been proved in another way. See van der Waerden, loc. cit.
10) A theorem due to H. Ha. See: E. Witt, Theorie der quadratischen Formen in beliebigen Körpern, Crelle's Journal, vol. 176 (1937), pp. 31-44. Satz 19.
11) As is well known, this follows from the sum-relation between p-invariants of A.
12) Cf. T. Nakayama and Y. Matsushima, Über die multiplikative Gruppe einer p-adischen Divisionsalgebra, Proc. Imp. Acad. Jap. vol. 19 (1944), pp. 622-628.
13) A subspace E is called non-isotropic if the restriction of f on E is non-degenerate, and a vector v is called non-isotropic if the straight line {v} is non-isotropic. See, for example, J. Dieudonné, loc. cit.
15) After leading the first manuscript of this paper which had been consisting of §§ 1 and 2 of the present paper, Mr. N. Iwahori informed to the author the following result: Let A be an algebra over the real number field with the center Z, and assume that the group of inner automorphisms of A gives rise to the whole rotation group of the real vector space A/Z, then A/Z has the dimension three. Our Proposition 7 was obtained in generalizing this result. Mr. N. Iwabori found a proof of this proposition, in case of characteristic zero, by a completely different argument.

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