zbMATH — the first resource for mathematics

Quaternions as reflexive skew fields. (English) Zbl 1222.12003
From the text: Throughout this text a skew field, or a sfield, is a ring with a unit element 1, in which every non zero element has an inverse. Given $$D$$ an sfield with centre $$F$$ and $$\sigma$$ an involution of $$D$$, i.e. a transformation $$x\mapsto \sigma(x)$$ of $$D$$ for which $$\sigma(x + y) = \sigma(x) + \sigma(y)$$, $$\sigma(xy) = \sigma(y)\sigma(x)$$ and $$\sigma^2(x) = x$$, for any $$x, y\in D$$.
Jean Dieudonné, in his work on the structure of unitary groups, says in [Sur les groupes classiques. (Paris: Hermann) (1948; Zbl 0037.01304), p. 72] that $$D$$ is a reflexive sfield if all the elements of $$F$$ are fixed by $$\sigma$$ and if $$x+\sigma(x)$$ and $$x\sigma(x)$$ belong to $$F$$, for any $$x\in D$$. He shows (by completing his proof in [Am. J. Math. 75, 665–678 (1953; Zbl 0051.01803)]) that non-commutative reflexive sfields are generalized quaternion sfields. His proof, amongst other things, relies on theorems concerning the structure of sfields of finite rank over their center, proved earlier this century in the 1920s. Now, Eliakim Hastings Moore had already stated and provided an elementary proof for the above-mentioned result provided by Dieudonné for sfields of characteristic other than 2. In 1922, E. H. Moore has defined a determinant for hermitian matrices with coefficients in those fields whose elements, fixed by the involution, are central [Bull. Am. Math. Soc. 28, 161–162 (1922; JFM 48.0128.07)].
Moore’s proof [General analysis. With the cooperation of Raymond Walter Barnard. Pt. I. Mem. Am. Philos. Soc. Vol. 1. Philadelphia: Am. Philos. Soc. (1935; Zbl 0013.11605 and JFM 61.0433.06), pp. 104–107] is based on an identity valid for any sfield with involution whose elements fixed by the involution are central. This identity, which was expressed by W. R. Hamilton himself [Elements of quaternions. (1866), p. 317; reprinted Cambridge (2009; Zbl 1204.01046)] for usual quaternions, is, in this case a translation of the identity $\langle\vec z\times\vec t, \vec y\rangle \vec x -\langle\vec z\times\vec t, \vec x\rangle \vec y - \langle\vec x\times\vec y, \vec t\rangle \vec z +\langle\vec x\times\vec y, \vec z\rangle \vec t = 0 \tag{1}$ in which $$x,y,z$$ and $$t$$ are vectors in usual three-dimensional space, and $$\langle\vec x, \vec y\rangle$$ and $$\langle\vec x\times\vec y\rangle$$ are, respectively, the scalar and vector products of $$x$$ and $$y$$. Here we sketch out Dieudonné’s non elementary proof, present Moore’s proof because the book cited above is difficult to obtain, and also present another elementary proof based on the fact that if $$V = \{x\in D\mid \sigma(x) = -x\}$$, then $$V$$ is a vector space on the subfield $$F_0$$ of the elements of $$F$$ fixed by $$\sigma$$ and the application $$x\mapsto x^2$$ is a regular quadratic form on that vector space.
MSC:
 12E15 Skew fields, division rings 16K20 Finite-dimensional division rings
Full Text:
References:
 [1] Dieudonné J., Sur les groupes classiques,Actualités scientifiques et industrielles, No. 1040, Paris, Hermann, 1948. · Zbl 0037.01304 [2] Dieudonné J., On the structure of unitary groups (II),Amer. J. Math.,75 p. 67, (1953). · Zbl 0051.01803 [3] Moore E. H., On the determinant of an hermitian matrix of quaternionic elements,Bull Amer. Math. Soc.,28 161–162 (1922). · JFM 48.0128.07 [4] Moore E. H., with the cooperation of K. W. Barnard, ”General analysis”, Part 1, Memoirs of the American Philosophical Society, Philadelphia, 1935. [5] Hunger Parshall Karen and David E. Rowe, ”The Emergence of the American Mathematical Research Community”, 1876–1900: Sylvester J. J., Felix Klein and E. H. Moore, American Mathematical Society, 1994. · Zbl 0802.01005 [6] Hamilton W. R., ”Elements of quaternions”, Longmans, Green, & co., London, 1866. · Zbl 1204.01046 [7] Herstein I. N., ”Topics in Algebra”, Xerox College Publishing, 1964. · Zbl 0122.01301 [8] Bourbaki N., ”Éléments de Mathématique”, Chap. 1 à 3, Hermann, Paris, 1970. [9] Knus M. A., A.S. Merkurjev, M. Rost and J.P. Tignol, The Book of Involution,Amer. Math. Soc. Coll. Pub.,44, 1998. · Zbl 0955.16001 [10] Cohn P. M., Skew Field Constructions, London Math. Soc.Lect. Notes Series No 27, Cambridge University Press, 1977. · Zbl 0355.16009 [11] Herstein I. N., Noncommutative rings, the carusMaths Monographs No 15, MAA, 1973. [12] Cohn P. M., ”Algebra”, Second Edition, vol. 2, John Wiley & sons, 1989. · Zbl 0703.00002 [13] Jacobson N.,Lectures in Abstract Algebra, III, Van Nostrand-Reinhold, 1964. · Zbl 0124.27002 [14] Morandi P. J., ”Lie Algebras, Composition Algebras, and the Existence of Cross Products on Finite-Dimensional Vector Space”, Expo. Math., à paraître. · Zbl 0976.17006 [15] Dyson F. J., Quaternion determinants,Helv. Phys. Acta 45, 289–302 (1972). [16] van Praag P., Sur la norme réduite du déterminant de Dieudonné des matrices quaternioniennes,J. Algebra 136, 265–274 (1991). · Zbl 0724.15013 · doi:10.1016/0021-8693(91)90047-C [17] Aslaksen H., Quaternionic Determinants,The Math. Intel. 18, No 3, 57–65 (1996). · Zbl 0881.15007 · doi:10.1007/BF03024312 [18] van Praag P., Les formes hermitiennes quaternioniennes et le déterminant d’Eliakim Hastings Moore,Bull. Soc. Math. Belgique, Sér. A42, 767–775 (1990). · Zbl 0731.15013 [19] Tignol J. P., Pfaffiens et Déterminant de E. H. Moore,Bull. Belgian Math. Soc., Simon Stevin,6, 537–539 (1999). [20] Lang S., ”Linear Algebra”, Second Edition, Addison-Wesley Pub. Company, 1971. [21] van Praag P., Une caractérisation des corps de quaternions,Bull. Soc. Math. Belgique,10, 283–285 (1968). · Zbl 0185.09003 [22] Jacobson N., Structure of rings,Amer. Math. Soc. Colloq. Pub.,XXXVII, (1956). · Zbl 0073.02002 [23] Cohn P. M.,Algebra, Second Edition, vol. 3, 1991. [24] van Praag P., Sur certains corps non commutatifs,Bull. Math. de la Soc. Sci. Math. de la R.S. de Roumanie, t. 14 (62), nr. 4, 451–453 (1970). · Zbl 0249.16010 [25] Tignol J. P.,Lettre à l’auteur, 25 août (1999).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.