Editing Hypercomplex number (section)

== History ==
In the nineteenth century, [[number system]]s called [[quaternion]]s, [[tessarine]]s, [[coquaternion]]s, [[biquaternion]]s, and [[octonion]]s became established concepts in mathematical literature, extending the real and [[complex number]]s. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them.

The cataloguing project began in 1872 when [[Benjamin Peirce]] first published his ''Linear Associative Algebra'', and was carried forward by his son [[Charles Sanders Peirce]].<ref>{{citation |title=Linear Associative Algebra |journal=[[American Journal of Mathematics]] |volume=4 |issue=1 |pages=221–6 |year=1881 |jstor=2369153|last1= Peirce|first1= Benjamin|doi=10.2307/2369153 |url=http://archive.org/details/linearassocalgeb00pierrich }}</ref>  Most significantly, they identified the [[nilpotent]] and the [[idempotent element (ring theory)|idempotent element]]s as useful hypercomplex numbers for classifications. The [[Cayley–Dickson construction]] used [[involution (mathematics)|involution]]s to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: [[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]] says finite-dimensional real [[composition algebra]]s are the reals <math>\mathbb{R}</math>, the complexes <math>\mathbb{C}</math>, the quaternions <math>\mathbb{H}</math>, and the octonions <math>\mathbb{O}</math>, and the [[Frobenius theorem (real division algebras)|Frobenius theorem]] says the only real [[associative division algebra]]s are <math>\mathbb{R}</math>, <math>\mathbb{C}</math>, and <math>\mathbb{H}</math>. In 1958 [[Frank Adams|J. Frank Adams]] published a further generalization in terms of Hopf invariants on ''H''-spaces which still limits the dimension to 1, 2, 4, or 8.<ref name="Adams1958">{{citation | jstor=1970147 | title=On the Non-Existence of Elements of Hopf Invariant One | author=Adams, J. F. | journal=Annals of Mathematics |date=July 1960  | volume=72 | issue=1 | pages=20–104 | doi=10.2307/1970147| url=http://www.math.rochester.edu/people/faculty/doug/otherpapers/Adams-HI1.pdf | citeseerx=10.1.1.299.4490 }}</ref>

It was [[matrix (mathematics)|matrix algebra]] that harnessed the hypercomplex systems. For instance, 2 x 2 [[real matrix|real matrices]] were found isomorphic to [[coquaternion]]s. Soon the matrix paradigm began to explain several others as they were represented by matrices and their operations.  In 1907 [[Joseph Wedderburn]] showed that associative hypercomplex systems could be represented by [[square matrices]], or [[direct product]]s of algebras of square matrices.<ref>{{citation |author=J.H.M. Wedderburn |author-link=Joseph Wedderburn | title=On Hypercomplex Numbers |journal=Proceedings of the London Mathematical Society |volume=6 | pages=77–118 |year=1908 | doi= 10.1112/plms/s2-6.1.77 |url=https://zenodo.org/record/1447798 }}</ref><ref>[[Emil Artin]] later generalized Wedderburn's result so it is known as the [[Artin–Wedderburn theorem]]</ref> From that date the preferred term for a ''hypercomplex system'' became ''[[associative algebra]]'', as seen in the title of Wedderburn's thesis at [[University of Edinburgh]]. Note however, that non-associative systems like octonions and  [[hyperbolic quaternion]]s represent another type of hypercomplex number.

As [[Thomas W. Hawkins Jr.|Thomas Hawkins]]<ref>{{citation |first=Thomas |last=Hawkins |title=Hypercomplex numbers, Lie groups, and the creation of group representation theory |journal=[[Archive for History of Exact Sciences]] |volume=8 |pages=243–287 |year=1972 |issue=4 |doi=10.1007/BF00328434 |s2cid=120562272 }}</ref> explains, the hypercomplex numbers are stepping stones to learning about [[Lie group]]s and [[group representation]] theory. For instance, in 1929 [[Emmy Noether]] wrote on "hypercomplex quantities and representation theory".<ref>{{citation | last = Noether | first = Emmy | year = 1929 | title = Hyperkomplexe Größen und Darstellungstheorie | trans-title = Hypercomplex Quantities and the Theory of Representations | journal = Mathematische Annalen | volume = 30 | pages = 641–92 | doi = 10.1007/BF01187794 | s2cid = 120464373 | language = de | url = http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002371448&L=1 | access-date = 2016-01-14 | archive-url = https://web.archive.org/web/20160329230805/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002371448&L=1 | archive-date = 2016-03-29 | url-status = dead }}</ref> In 1973 [[Isaiah Kantor|Kantor]] and Solodovnikov published a textbook on hypercomplex numbers which was translated in 1989.<ref name=KS78>Kantor, I.L., Solodownikow (1978), ''Hyperkomplexe Zahlen'', BSB B.G. Teubner Verlagsgesellschaft, Leipzig</ref><ref>{{Citation | last1=Kantor | first1=I. L. | last2=Solodovnikov | first2=A. S. | title=Hypercomplex numbers | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-96980-0 | mr=996029 | year=1989 | url-access=registration | url=https://archive.org/details/hypercomplexnumb0000kant }}</ref>

[[Karen Parshall]] has written a detailed exposition of the heyday of hypercomplex numbers,<ref>{{citation |author-link=Karen Parshall |first=Karen |last=Parshall |title=Joseph H. M. Wedderburn and the structure theory of algebras |journal=Archive for History of Exact Sciences |volume=32 |pages=223–349 |year=1985 |issue=3–4 |doi=10.1007/BF00348450 |s2cid=119888377 }}</ref> including the role of mathematicians including [[Theodor Molien]]<ref>{{citation |author-link=Theodor Molien |first=Theodor |last=Molien |title=Ueber Systeme höherer complexer Zahlen |journal=Mathematische Annalen |volume=41 |issue=1 |pages=83–156 |year=1893 |doi=10.1007/BF01443450 |s2cid=122333076 |url=https://zenodo.org/record/2029540}}</ref> and [[Eduard Study]].<ref>{{citation |author-link=Eduard Study |first=Eduard |last=Study |year=1898 |chapter=Theorie der gemeinen und höhern komplexen Grössen |title=[[Klein's encyclopedia|Encyclopädie der mathematischen Wissenschaften]] |volume=I A |issue=4 |pages=147–183}}</ref> For the transition to [[Abstract algebra|modern algebra]], [[Bartel van der Waerden]] devotes thirty pages to hypercomplex numbers in his ''History of Algebra''.<ref>{{citation |author-link=B.L. van der Waerden |first=B.L. |last=van der Waerden |year=1985 |title=A History of Algebra |chapter=10. The discovery of algebras, 11. Structure of algebras |publisher=Springer |isbn=3-540-13610X}}</ref>