Okubo algebra

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].

Following Okubo, [a7], let be the set of all traceless Hermitian matrices. The Okubo algebra is the vector space over the complex numbers spanned by the set with product defined by

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 .

The elements

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 Lie-admissible algebra; Algebra with associative powers); none of these algebras will have a unit element.

References

[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)