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
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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]:
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The elements (
) form an orthonormal basis; the multiplication follows from
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The constants and
must satisfy
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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 ![]() |
[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://encyclopediaofmath.org/index.php?title=Okubo_algebra&oldid=13007