Okubo algebra

Discovered by S. Okubo [a2] when searching for an algebraic structure to model $\operatorname { su } ( 3 )$ particle physics. Okubo looked for an algebra that is $8$-dimensional over the complex numbers, power-associative and, unlike the octonion algebra, has the Lie algebra $A _ { 2 }$ as both its derivation algebra and minus algebra. His algebra provides an important example of a division algebra that is $8$-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 $M$ be the set of all $3 \times 3$ traceless Hermitian matrices. The Okubo algebra $P _ { 8 }$ is the vector space over the complex numbers spanned by the set $M$ with product $*$ defined by

\begin{equation*} X ^ { * } Y = \mu X Y + \nu Y X + \frac { 1 } { 6 } \operatorname { Tr } ( X Y ), \end{equation*}

where $X Y$ denotes the usual matrix product of $X$ and $Y$, $\operatorname { Tr } ( X Y )$ is the trace of the matrix $X Y$ (cf. also Trace of a square matrix) and the constants $\mu$ and $\nu$ satisfy $3 \mu \nu = \mu + \nu = 1$, that is, $\mu = \overline { \nu } = ( 3 \pm i \sqrt { 3 } ) / 6$. In the discussion below, $\mu = ( 3 + i \sqrt { 3 } ) / 6$. The algebra $P _ { 8 }$ is not a division algebra; however, it contains a division algebra. The real vector space spanned by the set $M$ is a subring $\widetilde { P _ { 8 } }$ of $P _ { 8 }$ under the product $*$ and is a division algebra over the real numbers. Both the algebras $P _ { 8 }$ and $\widetilde { P _ { 8 } }$ are $8$-dimensional over their respective fields of scalars.

An explicit construction of the algebra $P _ { 8 }$ can be given in terms of the following basis of $3 \times 3$ traceless Hermitian matrices, introduced by M. Gell-Mann [a1]:

\begin{equation*} \lambda _ { 1 } = \left( \begin{array} { l l l } { 0 } & { 1 } & { 0 } \\ { 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } \end{array} \right), \lambda _ { 2 } = \left( \begin{array} { c c c } { 0 } & { - i } & { 0 } \\ { i } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } \end{array} \right), \end{equation*}

\begin{equation*} \lambda _ { 3 } = \left( \begin{array} { c c c } { 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } \end{array} \right) , \lambda _ { 4 } = \left( \begin{array} { c c c } { 0 } & { 0 } & { 1 } \\ { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 0 } \end{array} \right) , \lambda _ { 5 } = \left( \begin{array} { c c c } { 0 } & { 0 } & { - i } \\ { 0 } & { 0 } & { 0 } \\ { i } & { 0 } & { 0 } \end{array} \right) , \lambda _ { 6 } = \left( \begin{array} { c c c } { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 1 } \\ { 0 } & { 1 } & { 0 } \end{array} \right), \end{equation*}

\begin{equation*} \lambda _ { 7 } = \left( \begin{array} { c c c } { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { - i } \\ { 0 } & { i } & { 0 } \end{array} \right) , \lambda _ { 8 } = \left( \begin{array} { c c c } { \frac { 1 } { \sqrt { 3 } } } & { 0 } & { 0 } \\ { 0 } & { \frac { 1 } { \sqrt { 3 } } } & { 0 } \\ { 0 } & { 0 } & { \frac { - 2 } { \sqrt { 3 } } } \end{array} \right). \end{equation*}

The elements $e _ { j } = \sqrt { 3 } \lambda _ { j }$ ($j = 1 , \dots , 8$) form an orthonormal basis; the multiplication follows from

\begin{equation*} e _ { j } * e _ { k } = \sum _ { l = 1 } ^ { 8 } ( \sqrt { 3 } d _ { j k l } - f _ { j k l } ) e _ { l }. \end{equation*}

The constants $d_{ j k l}$ and $f _ { j k l }$ must satisfy

\begin{equation*} d _ { j k l } = \frac { 1 } { 4 } \operatorname { Tr } [ ( \gamma _ { j } \gamma _ { k } + \lambda _ { k } \lambda _ { j } ) \lambda _ { l } ], \end{equation*}

\begin{equation*} f _ { j k l } = \frac { - i } { 4 } \operatorname { Tr } [ ( \lambda _ { j } \lambda _ { k } - \lambda _ { k } \lambda _ { j } ) \lambda _ { l } ]. \end{equation*}

A partial tabulation of the values of $d_{ j k l}$ and $f _ { j k l }$ can be found in [a1].

The norm $\mathbf{N} ( X )$ of $X = \sum _ { j = 1 } ^ { 8 } X _ { j } e_j$ is ${\bf N} ( X ) = \sum _ { j = 1 } ^ { 8 } X _ { j } ^ { 2 }$. In the case of the algebra $\widetilde { P _ { 8 } }$, all the $X_j$ are real and $\mathbf{N} ( X ) = 0$ if and only if $X = 0$.

The elements

\begin{equation*} Y _ { j } = - \sqrt { 3 } \lambda _ { j } ( j = 1,2,3 ) , Y _ { 4 } = \sqrt { 3 } \lambda _ { 8 } \end{equation*}

generate a $4$-dimensional subalgebra, denoted by $P _ { 4 }$. Likewise, any non-identity element $\xi $ will generate a $2$-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.

References

[a1]	M. Gell–Mann, "Symmetries of baryons and mesons" Phys. Rev. , 125 (1962) pp. 1067–1084
[a2]	S. Okubo, "Pseudo-quaternion and pseudo-octonion algebras" Hadronic J. , 1 (1978) pp. 1250–1278. Zbl 0417.17011
[a3]	S. Okubo, "Deformation of the Lie-admissible pseudo-octonion algebra into the octonion algebra" Hadronic J. , 1 (1978) pp. 1383–1431. Zbl 0417.17012
[a4]	S. Okubo, "Octonion as traceless $3 \times 3$ matrices via a flexible Lie-admissible algebra" Hadronic J. , 1 (1978) pp. 1432–1465. Zbl 0417.17013
[a5]	S. Okubo, "A generalization of Hurwitz theorem and flexible Lie-admissible algebras" Hadronic J. , 3 (1978) pp. 1–52. Zbl 0418.17004
[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)