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Discovered by S. Okubo [[#References|[a2]]] when searching for an algebraic structure to model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o1300302.png" /> particle physics. Okubo looked for an algebra that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o1300303.png" />-dimensional over the complex numbers, power-associative and, unlike the [[octonion]] algebra, has the [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o1300304.png" /> as both its derivation algebra and minus algebra. His algebra provides an important example of a [[Division algebra|division algebra]] that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o1300305.png" />-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 [[#References|[a3]]], [[#References|[a5]]], [[#References|[a4]]], [[#References|[a7]]], and [[#References|[a6]]].
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Following Okubo, [[#References|[a7]]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o1300306.png" /> be the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o1300307.png" /> traceless Hermitian matrices. The Okubo algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o1300308.png" /> is the [[Vector space|vector space]] over the complex numbers spanned by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o1300309.png" /> with product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003010.png" /> defined by
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003011.png" /></td> </tr></table>
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Discovered by S. Okubo [[#References|[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|Lie algebra]] $A _ { 2 }$ as both its derivation algebra and minus algebra. His algebra provides an important example of a [[Division algebra|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 [[#References|[a3]]], [[#References|[a5]]], [[#References|[a4]]], [[#References|[a7]]], and [[#References|[a6]]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003012.png" /> denotes the usual matrix product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003015.png" /> is the trace of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003016.png" /> (cf. also [[Trace of a square matrix|Trace of a square matrix]]) and the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003018.png" /> satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003019.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003020.png" />. In the discussion below, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003021.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003022.png" /> is not a division algebra; however, it contains a division algebra. The real vector space spanned by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003023.png" /> is a subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003025.png" /> under the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003026.png" /> and is a division algebra over the real numbers. Both the algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003028.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003029.png" />-dimensional over their respective fields of scalars.
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Following Okubo, [[#References|[a7]]], let $M$ be the set of all $3 \times 3$ traceless Hermitian matrices. The Okubo algebra $P _ { 8 }$ is the [[Vector space|vector space]] over the complex numbers spanned by the set $M$ with product $*$ defined by
  
An explicit construction of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003030.png" /> can be given in terms of the following basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003031.png" /> traceless Hermitian matrices, introduced by M. Gell-Mann [[#References|[a1]]]:
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\begin{equation*} X ^ { * } Y = \mu X Y + \nu Y X + \frac { 1 } { 6 } \operatorname { Tr } ( X Y ), \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003032.png" /></td> </tr></table>
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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|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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003033.png" /></td> </tr></table>
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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 [[#References|[a1]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003034.png" /></td> </tr></table>
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\begin{equation*} \lambda _ { 1 } = \left( \begin{array} { l l l } { 0 } &amp; { 1 } &amp; { 0 } \\ { 1 } &amp; { 0 } &amp; { 0 } \\ { 0 } &amp; { 0 } &amp; { 0 } \end{array} \right), \lambda _ { 2 } = \left( \begin{array} { c c c } { 0 } &amp; { - i } &amp; { 0 } \\ { i } &amp; { 0 } &amp; { 0 } \\ { 0 } &amp; { 0 } &amp; { 0 } \end{array} \right), \end{equation*}
  
The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003035.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003036.png" />) form an orthonormal basis; the multiplication follows from
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\begin{equation*} \lambda _ { 3 } = \left( \begin{array} { c c c } { 1 } &amp; { 0 } &amp; { 0 } \\ { 0 } &amp; { - 1 } &amp; { 0 } \\ { 0 } &amp; { 0 } &amp; { 0 } \end{array} \right) , \lambda _ { 4 } = \left( \begin{array} { c c c } { 0 } &amp; { 0 } &amp; { 1 } \\ { 0 } &amp; { 0 } &amp; { 0 } \\ { 1 } &amp; { 0 } &amp; { 0 } \end{array} \right) , \lambda _ { 5 } = \left( \begin{array} { c c c } { 0 } &amp; { 0 } &amp; { - i } \\ { 0 } &amp; { 0 } &amp; { 0 } \\ { i } &amp; { 0 } &amp; { 0 } \end{array} \right) , \lambda _ { 6 } = \left( \begin{array} { c c c } { 0 } &amp; { 0 } &amp; { 0 } \\ { 0 } &amp; { 0 } &amp; { 1 } \\ { 0 } &amp; { 1 } &amp; { 0 } \end{array} \right), \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003037.png" /></td> </tr></table>
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\begin{equation*} \lambda _ { 7 } = \left( \begin{array} { c c c } { 0 } &amp; { 0 } &amp; { 0 } \\ { 0 } &amp; { 0 } &amp; { - i } \\ { 0 } &amp; { i } &amp; { 0 } \end{array} \right) , \lambda _ { 8 } = \left( \begin{array} { c c c } { \frac { 1 } { \sqrt { 3 } } } &amp; { 0 } &amp; { 0 } \\ { 0 } &amp; { \frac { 1 } { \sqrt { 3 } } } &amp; { 0 } \\ { 0 } &amp; { 0 } &amp; { \frac { - 2 } { \sqrt { 3 } } } \end{array} \right). \end{equation*}
  
The constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003039.png" /> must satisfy
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The elements $e _ { j } = \sqrt { 3 } \lambda _ { j }$ ($j = 1 , \dots , 8$) form an orthonormal basis; the multiplication follows from
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003040.png" /></td> </tr></table>
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\begin{equation*} e _ { j } * e _ { k } = \sum _ { l = 1 } ^ { 8 } ( \sqrt { 3 } d _ { j k l } - f _ { j k l } ) e _ { l }. \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003041.png" /></td> </tr></table>
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The constants $d_{ j k l}$ and $f _ { j k l }$ must satisfy
  
A partial tabulation of the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003043.png" /> can be found in [[#References|[a1]]].
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\begin{equation*} d _ { j k l } = \frac { 1 } { 4 } \operatorname { Tr } [ ( \gamma _ { j } \gamma _ { k } + \lambda _ { k } \lambda _ { j } ) \lambda _ { l } ], \end{equation*}
  
The norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003045.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003046.png" />. In the case of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003047.png" />, all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003048.png" /> are real and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003049.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003050.png" />.
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\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 [[#References|[a1]]].
 +
 
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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
 
The elements
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003051.png" /></td> </tr></table>
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\begin{equation*} Y _ { j } = - \sqrt { 3 } \lambda _ { j } ( j = 1,2,3 ) , Y _ { 4 } = \sqrt { 3 } \lambda _ { 8 } \end{equation*}
  
generate a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003052.png" />-dimensional subalgebra, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003053.png" />. Likewise, any non-identity element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003054.png" /> will generate a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003055.png" />-dimensional subalgebra.
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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 [[Lie-admissible algebra|Lie-admissible algebra]]; [[Algebra with associative powers|Algebra with associative powers]]); none of these algebras will have a unit element.
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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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Gell–Mann,  "Symmetries of baryons and mesons"  ''Phys. Rev.'' , '''125'''  (1962)  pp. 1067–1084</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Okubo,  "Pseudo-quaternion and psuedo-octonion algebras"  ''Hadronic J.'' , '''1'''  (1978)  pp. 1250–1278</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Okubo,  "Deformation of the Lie-admissible pseudo-octonion algebra into the octonion algebra"  ''Hadronic J.'' , '''1'''  (1978)  pp. 1383–1431</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Okubo,  "Octonion as traceless <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130030/o13003056.png" /> matrices via a flexible Lie-admissible algebra"  ''Hadronic J.'' , '''1'''  (1978)  pp. 1432–1465</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Okubo,  "A generalization of Hurwitz theorem and flexible Lie-admissible algebras"  ''Hadronic J.'' , '''3'''  (1978)  pp. 1–52</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Okubo,  H.C. Myung,  "Some new classes of division algebras"  ''J. Algebra'' , '''67'''  (1980)  pp. 479–490</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Okubo,  "Introduction to octonion and other non-associative algebras in physics" , Cambridge Univ. Press  (1995)</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  M. Gell–Mann,  "Symmetries of baryons and mesons"  ''Phys. Rev.'' , '''125'''  (1962)  pp. 1067–1084</td></tr>
 +
<tr><td valign="top">[a2]</td> <td valign="top">  S. Okubo,  "Pseudo-quaternion and pseudo-octonion algebras"  ''Hadronic J.'' , '''1'''  (1978)  pp. 1250–1278. {{ZBL|0417.17011}}</td></tr>
 +
<tr><td valign="top">[a3]</td> <td valign="top">  S. Okubo,  "Deformation of the Lie-admissible pseudo-octonion algebra into the octonion algebra"  ''Hadronic J.'' , '''1'''  (1978)  pp. 1383–1431. {{ZBL|0417.17012}}</td></tr>
 +
<tr><td valign="top">[a4]</td> <td valign="top">  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}}</td></tr>
 +
<tr><td valign="top">[a5]</td> <td valign="top">  S. Okubo,  "A generalization of Hurwitz theorem and flexible Lie-admissible algebras"  ''Hadronic J.'' , '''3'''  (1978)  pp. 1–52.  {{ZBL|0418.17004}}</td></tr>
 +
<tr><td valign="top">[a6]</td> <td valign="top">  S. Okubo,  H.C. Myung,  "Some new classes of division algebras"  ''J. Algebra'' , '''67'''  (1980)  pp. 479–490</td></tr>
 +
<tr><td valign="top">[a7]</td> <td valign="top">  S. Okubo,  "Introduction to octonion and other non-associative algebras in physics" , Cambridge Univ. Press  (1995)</td></tr>
 +
</table>

Latest revision as of 10:21, 8 March 2021

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)
How to Cite This Entry:
Okubo algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Okubo_algebra&oldid=30357
This article was adapted from an original article by G.P. Wene (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article