Biquaternium

Biquaternium^[1] (Anglice biquaternion) in algebra abstracta est ullus ex numeris $w+xi+yj+zk$ , ubi $w,\ x,\ y$ , et $z$ sunt numeri complexi vel eorum variantes, et elementa $1,\ i,\ j,\ k$ se multiplicant ut in grege quaterniorum et cum eorum coefficientibus commutantur. Sunt tria biquaterniorum genera, quae cum numeris complexis eorumque variantibus congruunt:

Biquaternia cum coefficientes numeri complexi sint
Biquaternia fissa cum coefficientes numeri complexi et fissi sint
Quaternia binaria cum coefficients numeri binarii sint.

Biquaternia ordinaria, a Gulielmo Rowan Hamilton excogitata, annis 1844 et 1850 in Proceedings of the Royal Irish Academy divulgata sunt.^[2] Inter gravissimos huius generis biquaterniorum suasores numerantur Arthurus W. Conway, Cornelius Lanczos, Alexander Macfarlane, Ludovicus Silberstein. Quasi-sphaerium, unitas biquaterniorum, gregem Lorentzianum repraesentat, qui fundamenta relativitatis specialis est.

Algebra biquaterniorum putari potest productus tensorum $\mathbb {C} \otimes \mathbb {H}$ (supra reales) ubi $\mathbb {C}$ est campus numerorum complexorum et $\mathbb {H}$ est algebra divisionis quaterniorum realium; quod breviter significat biquaternia esse complexificationem tantum quaterniorum. Biquaternia, cum genus algebrae complexae videantur, isomorphica ad algebram $2\times 2$ matricum complexarum $M_{2}(\mathbb {C} )$ sunt. Etiam sunt isomorphica ad nonnullas algebras Cliffordianas, inter quas $1=\mathbb {H} (\mathbb {C} )=C_{3}^{0}(\mathbb {C} )=C\ell _{2}(\mathbb {C} )=C\ell _{1,2}(\mathbb {R} )$ ,^[3] algebram Paulianam $C\ell _{3,0}(\mathbb {R} )$ ,^[3]^[4] et par algebrae spatiotemporalis pars $C\ell _{1,3}^{0}(\mathbb {R} )=C\ell _{3,1}^{0}(\mathbb {\mathbb {R} } )$ .^[4]

Notae

↑ Fons nominis Latini desideratur (addito fonte, hanc formulam remove)
↑ Proceedings of the Royal Irish Academy November 1844 (NA) et 1850: 388 in Google Books.
↑ ^3.0 ^3.1 D. J. H. Garling, Clifford Algebras: An Introduction (Cantabrigiae: Cambridge University Press, 2011).
↑ ^4.0 ^4.1 Francis and Kosowsky (2005), "The construction of spinors in geometric algebra," Annals of Physics 317: 384–409. Nexus ad commentarium.

Bibliographia

Lege de quaterniis in Vicilibris.

Buchheim, Arthur. 1885. "A Memoir on biquaternions." American Journal of Mathematics 7 (4): 293–326. JSTOR.
Conway, Arthur W. 1911. "On the application of quaternions to some recent developments in electrical theory." Proceedings of the Royal Irish Academy 29A: 1–9.
Furey, C. 2012. "Unified Theory of Ideals." Physics Review D 86 (2): 025024. doi:10.1103/PhysRevD.86.025024. Bibcode 2012PhRvD..86b5024F |s2cid=118458623.
Girard, P. R.1984. "The quaternion group and modern physics." European Journal of Physics 5 (1): 25–32. Bibcode 1984EJPh....5...25G. doi:10.1088/0143-0807/5/1/007.
Hamilton, William Rowan. 1866. Elements of Quaternions. Ed. a Gulielmo Eduino Hamilton, filio patris mortui. University of Dublin Press.
Hamilton, William Rowan. 1899. Elements of Quaternions, vol. I, (1901). vol. II. Liber editus a Carolo Jasper Joly. Longmans, Green & Co.
Kilmister, C. W. 1994. Eddington's search for a fundamental theory. Cantabrigiae: Cambridge University Press. ISBN 978-0-521-37165-0.
Kravchenko, Vladislav. 2003. Applied Quaternionic Analysis. Heldermann Verlag. ISBN 3-88538-228-8.
Sangwine, Stephen J., Todd A. Ell, et Nicolas Le Bihan. 2010. "Fundamental representations and algebraic properties of biquaternions or complexified quaternions." Advances in Applied Clifford Algebras 21 (3): 1–30. doi:10.1007/s00006-010-0263-3. Arxiv 1001.0240.
Sangwine, Stephen J., et Daniel Alfsmann. 2010. "Determination of the biquaternion divisors of zero, including idempotents and nilpotents." Advances in Applied Clifford Algebras 20 (2): 401–410. Bibcode 2008arXiv0812.1102S. doi:10.1007/s00006-010-0202-3.
Silberstein, Ludwik. 1912. "Quaternionic form of relativity." Philosophical Magazine series 6: 23 (137): 790–809. doi:10.1080/14786440508637276. URL.
Silberstein, Ludwik. 1914. The Theory of Relativity.
Synge, J. L. 1972. "Quaternions, Lorentz transformations, and the Conway-Dirac-Eddington matrices." Communications of the Dublin Institute for Advanced Studies series A, vol. 21.
Tanişli, M. 2006. "Gauge transformation and electromagnetism with biquaternions." Europhysics Letters 74 (4): 569. Bibcode = 2006EL.....74..569T. doi:10.1209/epl/i2005-10571-6 .

[fons-nominis-desideratus-1] Fons nominis Latini desideratur (addito fonte, hanc formulam remove)

[2] Proceedings of the Royal Irish Academy November 1844 (NA) et 1850: 388 in Google Books.

[Garling-3] 3.0 ^3.1 D. J. H. Garling, Clifford Algebras: An Introduction (Cantabrigiae: Cambridge University Press, 2011).

[FrancisKosowsky-4] 4.0 ^4.1 Francis and Kosowsky (2005), "The construction of spinors in geometric algebra," Annals of Physics 317: 384–409. Nexus ad commentarium.

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