Discovered by S. Okubo [a2] when searching for an algebraic structure to model particle physics. Okubo looked for an algebra that is -dimensional over the complex numbers, power-associative and, unlike the octonion algebra, has the Lie algebra as both its derivation algebra and minus algebra. His algebra provides an important example of a division algebra that is -dimensional over the real numbers with a norm permitting composition that is not alternative. For more information on these algebras, their generalizations and the physics, see [a3], [a5], [a4], [a7], and [a6].
where denotes the usual matrix product of and , is the trace of the matrix (cf. also Trace of a square matrix) and the constants and satisfy , that is, . In the discussion below, . The algebra is not a division algebra; however, it contains a division algebra. The real vector space spanned by the set is a subring of under the product and is a division algebra over the real numbers. Both the algebras and are -dimensional over their respective fields of scalars.
An explicit construction of the algebra can be given in terms of the following basis of traceless Hermitian matrices, introduced by M. Gell-Mann [a1]:
The elements () form an orthonormal basis; the multiplication follows from
The constants and must satisfy
A partial tabulation of the values of and can be found in [a1].
The norm of is . In the case of the algebra , all the are real and if and only if .
generate a -dimensional subalgebra, denoted by . Likewise, any non-identity element will generate a -dimensional subalgebra.
In addition to the above properties, each algebra will be flexible, power associative and Lie-admissible (cf. also Flexible identity; Lie-admissible algebra; Algebra with associative powers); none of these algebras will have a unit element.
|[a1]||M. Gell–Mann, "Symmetries of baryons and mesons" Phys. Rev. , 125 (1962) pp. 1067–1084|
|[a2]||S. Okubo, "Pseudo-quaternion and psuedo-octonion algebras" Hadronic J. , 1 (1978) pp. 1250–1278|
|[a3]||S. Okubo, "Deformation of the Lie-admissible pseudo-octonion algebra into the octonion algebra" Hadronic J. , 1 (1978) pp. 1383–1431|
|[a4]||S. Okubo, "Octonion as traceless matrices via a flexible Lie-admissible algebra" Hadronic J. , 1 (1978) pp. 1432–1465|
|[a5]||S. Okubo, "A generalization of Hurwitz theorem and flexible Lie-admissible algebras" Hadronic J. , 3 (1978) pp. 1–52|
|[a6]||S. Okubo, H.C. Myung, "Some new classes of division algebras" J. Algebra , 67 (1980) pp. 479–490|
|[a7]||S. Okubo, "Introduction to octonion and other non-associative algebras in physics" , Cambridge Univ. Press (1995)|
Okubo algebra. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Okubo_algebra&oldid=37374