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Difference between revisions of "Cayley numbers"

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Hypercomplex numbers (cf. [[Hypercomplex number|Hypercomplex number]]), namely, the elements of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021070/c0210701.png" />-dimensional algebra over the field of real numbers (the Cayley algebra). They were first considered by A. Cayley. The Cayley algebra may be derived via the Cayley–Dickson process from the algebra of quaternions (see [[Cayley–Dickson algebra|Cayley–Dickson algebra]]; [[Quaternion|Quaternion]]). It is the only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021070/c0210702.png" />-dimensional real alternative algebra without zero divisors (see [[Frobenius theorem|Frobenius theorem]]). The Cayley algebra is an algebra with unique division and with an identity; it is alternative, non-associative and non-commutative.
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Hypercomplex numbers (cf. [[Hypercomplex number|Hypercomplex number]]), namely, the elements of the 8-dimensional algebra over the field of real numbers (the Cayley algebra). They were first considered by A. Cayley. The Cayley algebra may be derived via the Cayley–Dickson process from the algebra of quaternions (see [[Cayley–Dickson algebra|Cayley–Dickson algebra]]; [[Quaternion|Quaternion]]). It is the only 8-dimensional real alternative algebra without zero divisors (see [[Frobenius theorem|Frobenius theorem]]). The Cayley algebra is an algebra with unique division and with an identity; it is alternative, non-associative and non-commutative.
  
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>
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|valign="top"|{{Ref|Ku}}|| valign="top"| A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)
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Revision as of 06:26, 22 April 2012

Hypercomplex numbers (cf. Hypercomplex number), namely, the elements of the 8-dimensional algebra over the field of real numbers (the Cayley algebra). They were first considered by A. Cayley. The Cayley algebra may be derived via the Cayley–Dickson process from the algebra of quaternions (see Cayley–Dickson algebra; Quaternion). It is the only 8-dimensional real alternative algebra without zero divisors (see Frobenius theorem). The Cayley algebra is an algebra with unique division and with an identity; it is alternative, non-associative and non-commutative.

References

[Ku] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
How to Cite This Entry:
Cayley numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cayley_numbers&oldid=17787
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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