Biquaternium

Biquaternium^[1] (Anglice biquaternion) in algebra abstracta est ullus ex numeris ${\displaystyle w+xi+yj+zk}$, ubi ${\displaystyle w,\ x,\ y}$, et ${\displaystyle z}$ sunt numeri complexi vel eorum variantes, et elementa ${\displaystyle 1,\ i,\ j,\ k}$ 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 ${\displaystyle \mathbb {C} \otimes \mathbb {H} }$ (supra reales) ubi ${\displaystyle \mathbb {C} }$ est campus numerorum complexorum et ${\displaystyle \mathbb {H} }$ est algebra divisionis quaterniorum realium; quod breviter significat biquaternia esse complexificationem tantum quaterniorum. Biquaternia, cum genus algebrae complexae videantur, isomorphica ad algebram ${\displaystyle 2\times 2}$ matricum complexarum ${\displaystyle M_{2}(\mathbb {C} )}$ sunt. Etiam sunt isomorphica ad nonnullas algebras Cliffordianas, inter quas ${\displaystyle 1=\mathbb {H} (\mathbb {C} )=C_{3}^{0}(\mathbb {C} )=C\ell _{2}(\mathbb {C} )=C\ell _{1,2}(\mathbb {R} )}$,^[3] algebram Paulianam ${\displaystyle C\ell _{3,0}(\mathbb {R} )}$,^[3][4] et par algebrae spatiotemporalis pars ${\displaystyle C\ell _{1,3}^{0}(\mathbb {R} )=C\ell _{3,1}^{0}(\mathbb {\mathbb {R} } )}$.^[4]

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 .