10

数理解析研究所講究録
第2008巻 2016年 10-20

A generaliged automorphism group andtriality groups for nonassociative algebras(1930‐2015)

In memory of Professor Susumu Okubo

Noriaki Kamiya1 and Susumu Okubo2

iDepartment of Mathematics, University of Aizu,
Aizuwakamatsu, Japan

2Department of Physics and Astronomy, University of Rochester,Rochester, \mathrm{N}\cdot.\mathrm{Y} U.S.A,

Abstract
In this paper, we introduce for triality groups and study examples of the group.

AMS classification, 17\mathrm{A}40, 22\mathrm{F}05,17\mathrm{B}60.17\mathrm{B}40.

Keywords, Symmetric composition algebras, triality groups, local and global triality
relation.

(1) Triality GroupLet A be an algebra (do not need the associativity) over a field F of charachteristic

not 2 with a bi‐linear product denoted by juxtaposition xy for x, y\in A.

Suppose that

triple g=(g_{1}, g_{2}, g_{3})\in(EpiA)^{3}
equipped with g_{j}(x+y)=g_{j}x+g_{\dot{}}y and g_{j}( $\alpha$ x)= $\alpha$ g_{j}(x) $\alpha$\in F satisfies a global trialitya,where EpiA denote the set of epimorphisms of A,,

relationg_{\mathrm{j}}(xy)=(g_{j+1}x)(g_{j+2}y)(1.1)

for any x, y\in A and for any j=1 2, 3, Here the index j is defined by modulo 3 so that,g_{j\pm 3}=g_{j}For the second triple

(1.2)

.

g^{J}=(g_{1}^{J}, g_{2}^{J}, g_{3}^{J})\in(Epi\mathcal{A})^{3} satisfying the same triality relation, we

introduce their product componetwise by

gg^{J}=(g_{1}g_{\acute{1} , g_{2}g_{2}^{J}, g_{3}g_{3}^{J})

.

(1.3)

Then, a set consisting of all such triples forms a group whicb we call the triality group of
A and write,

Trig (A)=\{g=(g_{1},g_{2},g_{3})\in(EpiA)^{3}|g_{j}(xy)=(g_{j+1}x)(g_{j+2}y), \forall x, y\in A, \forall j=1, 2, 3\}.

(1.4)

11

Here we emphasize that this group is clearly a generalization of the automorphism groupdefined byAuto (A)=\{g\in EpiA|g(xy)=(gx)(gy), \forall x, y\in A\}

(1.5)

.

Trig (A) is invariant under actions of an alternative group A_{4} (orequivalently the tetrahedral group) as follows. First, let $\phi$\in End(\mathrm{T}\mathrm{r}\mathrm{i}\mathrm{g}(A) by
We then note that$\phi$:g_{1}\rightarrow g_{2}\rightarrow g_{3}\rightarrow g_{1}(1.6)

which satisfies $\phi$^{3}=id and leaves Eq.(l.l) invariant.

Tus, Trig (A) is invariant underactions of the cyclic group Z3 generated by $\phi$.

We next introduce$\mu$=1 2, 3 by$\tau$_{ $\mu$}\in End( TrigA) for,$\tau$_{1}:g_{1}\rightarrow g_{1}, g_{2}\rightarrow-g_{2}, g_{3}\rightarrow-g_{3}$\tau$_{2}:g_{1}\rightarrow-g_{1}, g_{2}\rightarrow g_{2}, g_{3}\rightarrow-g_{3}(1.7)$\tau$_{3}:g_{1}\rightarrow-g_{1}, g_{2}\rightarrow-g_{2}, g_{3}\rightarrow g_{3},which leave Eq.(l.l) invariant again.

Moreover, tbey satisfy$\tau$_{ $\mu$}$\tau$_{ $\nu$}=$\tau$_{ $\nu$}$\tau$_{ $\mu$}, $\tau$_{ $\mu$}^{2}=id, $\tau$_{1}$\tau$_{2}$\tau$_{3}=id(1.8)

for $\mu$, \mathrm{v}=1 2, 3, so that (id, $\tau$_{1}, $\tau$_{2}, $\tau$_{3}) is isomorphic to the Kleins 4‐group K_{4}.
Further we note,$\phi \tau$_{ $\mu$}$\phi$^{-1}=$\tau$_{ $\mu$+1} (with $\tau$_{4}=$\tau$_{1} ).

(1.9)

,

Since Z3 and K_{4} generate the alternative group A_{4} this shows that A_{4} is an invariant,

group of Trig (A)If A is invalntive with the involution map x\rightarrow\overline{x} satisfying.

x==x, \overline{xy}=\overline{y}\overline{x}we(1.10),

define \overline{Q}\in End(A) for any Q\in End(A) by\overline{Qx}=\overline{Q}\overline{x}(1.11).

then, taking the.involution of Eq.(1.10) and letting x\leftrightar ow\overline{y}\overline{g}_{j}(xy)=(\overline{g}_{j+2}x)(\overline{g}_{j+1}y)
so

,

we

find

(1.12),

that $\theta$\in End(\mathrm{T}\mathrm{r}\mathrm{i}\mathrm{g}(A) defined by$\theta$ : g_{1}\rightarrow\overline{g}_{2}, g_{2}\rightarrow\overline{g}_{1}, g_{3}\rightarrow\overline{g}_{3}(1.13)

yields also a invariant operation of Trig (A) Moreover, we obtain.

$\phi \theta \phi$= $\theta,\ \theta$^{2}=id, $\theta \tau$_{1}$\theta$^{-1}=$\tau$_{2}, $\theta \tau$_{2}$\theta$^{-1}=$\tau$_{1}, $\theta \tau$_{3}$\theta$^{-1}=$\tau$_{3}.

(1.14)

Then, A_{4} togehter with $\theta$ give the S_{4} ‐symmetry for Trig (A) with identification of$\phi$=(1,2,3) $\tau$_{1}=(2,3)(1,4)$\tau$_{2}=(3.1)(2.4) $\tau$_{3}=(1,2)(3,4) $\theta$=(1,2),

in terms of the transpositions of the S_{4} ‐group.

,,(1.15)

12

Before going into further discussion, we note that Txig (A) for any A always contains

Kleins 4‐group K_{4} Let Id\in Epi(A) be defined by (Id)x=x for x\in A and set

a

,

,

\tilde{ $\tau$}_{0}= ( Id, Id, Id), \tilde{ $\tau$}_{1}= ( Id .‐Id, −Id), \tilde{ $\tau$}_{2}=(-Id, Id, -Id) \tilde{ $\tau$}_{3}= ( -Id −Id, Id). (1.16),,

Then, these 4 element of (Epi(A)^{3}) satisfy Eq.(1.8) in view of Eq.(1.3), so thàt they give
another Kleins 4‐group. Moreover, they satisfy \mathrm{E}\mathrm{q}.(1.1) Similarly, we have Trig (F)=
K_{4} for the simplest case of A=F.
In contrast to the global triality relation \mathrm{E}\mathrm{q}.(1.1) we may also consider the local
triality relation.

,,/t_{j}(xy)=(t_{j+1}x)y+x(t_{j+2}y)
for t_{j}\in End(A) with t_{j\pm 3}=t_{j}soLrt(1.17)

Analogously to Eq.(1.4), we introduce.

(A)=\{t=(t_{1}, t_{2}, t_{3})\in (End A)^{3}|t_{j}(xy)=(t_{j+1}x)y+x(t_{j+2}y), \forall x, y\in A, \forall i=1, 2, 3\}.
(1.18)

Then, it defines a Lie algebra now with component‐wise commutation relation (]K‐O,15]).
Here, so Lrt(A) stands for symmetric Lie‐related triple, which has been reformed to as

stri (A) in [0.05] instead. If we set

t_{\acute{j} =\displaystyle \sum_{k=1}^{3}$\alpha$_{j-k}t_{k}, (j=1,2,3)(1.19)

for any $\alpha$_{j}\in F satisfying $\alpha$_{\dot{}\pm 3}=$\alpha$_{j} then it is easy to verify that we have also,t'=(t_{1}', t_{2}', t_{3}')\in s\mathrm{o}Lrt(A).

(1.20)

If the exponential map t_{j}\rightarrow$\xi$_{j} is given by$\xi$_{j}=\displaystyle \exp(t_{i})=\sum_{n=0}^{\infty}\frac{1}{n!}(t_{j})^{n},(1.21)

is well defined, then we have shown in [K‐O.15J that the vality of$\xi$_{j}(xy)=($\xi$_{j+1}x)($\xi$_{j+2}y)(1.22)

provided with respect to t=(t_{1}, t_{2}, t_{3})\in s\mathrm{o}Lrt(At and vice‐versa.

Note that the existence ot the exponantial map requires the underlying field F to beat least of zero charachteristic.

We next introduce multiplication operators L(x) , R(x)\in End A by

L(x)y=xyR(x)y=yx
asusual.

(1.23)

Then, Eq.(1.17) yields

t_{j}L(x)=L(x)t_{j+2}+L(t_{\mathrm{j}+1}x) , (1.24a)t_{j}R(y)=R(y)t_{j+1}+R(t_{j+2}y) (1.24b)
while Eq.(l.l) gives

g_{j}L(x)=L(g_{j+1}x)g_{;i+2}, (1.25a)13

g_{\dot{}}R(y)=R(g_{j+2}y)g_{j+1}
any g=(g_{1}, g_{2}, g_{3})\in \mathrm{T}\mathrm{r}\mathrm{i}\mathrm{g}(A)(1.25b).

From Eqs.(1.25), we findg_{j}L(x)R(y)g_{j}^{-1}=L(g_{j+1}x)R(g_{j+1}y) (1.26a)g_{j}R(y)L(x)g_{j}^{-1}=R(g_{j+2}y)L(g_{\mathrm{j}+2}x)(1.26b).

We next introduce the notion of a regular triality algebra.
Def.1.1

Let d_{j}(x, y)\in End(A) for x, y\in A and for j=1 2, 3 be to satisfy,(i)d_{1}(x, y)=R(y)L(x)-R(x)L(y) , (1.27a)d_{2}(x, y)=L(y)R(x)-L(x)R(y)(ii)(1.27b).

The explicit form for d_{3}(x, y) is unspecified except ford_{3}(y, x)=-d_{3}(x, y) (1.27c)(iii)(d_{[}(x, y), d_{2}(x, y), d_{3}(x, y))\in s\mathrm{o}Lrt(A) i.e., they satisfy,d_{j}(x, y)(uv)=(d_{j+1}(x, y)u)v+u(d_{j+2}(x, y)v)

(1.28)

for any x, y, u, v\in A and for any j=1 2, 3. here the index over j is defined modulo 3 as
before.
,We call the algebra A satisfying these condition to be a regular triality algebra.

\mathrm{O}.15])([K‐

Def. 1.2
Condition (B): We have AA=A.
Condition (C): If some b\in A satisfies either L(b)=0,or

R(b)=0 then b=0.
,

We can now prove.

Proposition 1.3
Let A be a regular triality algebra satisfying either the condition (B) or (C).

We thenobtain the followings:

For any t=(t_{1}, t_{2}, t_{3})\in s\mathrm{o}Lrt(A) ,

we

jave

[t_{j}, d_{k}(x, y)]=d_{k}(t_{j-k}x, y)+d_{k}(x, t_{j-k}y)
Especially, if we choose t_{j}=d_{j}(u, v),

it yields also[d_{j}(u, v), d_{k}(x, y)]=d_{k}(d_{j-k}(u, v)x, y)+d_{k}(x, d_{j-k}(u, v)y)(ii)

(1^{\cdot}.29a).

.

(1.29b)For any g=(g_{1}, g_{2}, g_{3})\in \mathrm{T}\mathrm{r}\mathrm{i}\mathrm{g}(A) , we haveg_{j}d_{k}(x, y)g_{j}^{-1}=d_{k}(g_{j-k}x, y)+d_{k}(x, g_{j-k}y)(1.30)

14

for any j, k=1 2, 3 and for any u, v, x, y\in A,
,

‐

Proof

Since Eqs.(1.29) have been already proved in [K‐O,15], we will give only a proof ofEq.(1.30) below. In view of Eq.(1.26) and (1.27), we see that Eq.(1.30) holds valid forany j=1 2, 3 and for k=1 2. Therefore, it suffices to prove of the case of k=3
end, we set,,

D_{j,k} :\equiv g_{j}d_{k}(x, y)g_{j}^{-1}-d_{k}(s_{j-k}x, y)-d_{k}(x, g_{j-k}y).

To this(1.31)

a fixed x, y\in A
Then, as we have noted, we have D_{j.1}=D_{j.2}=0 identically.
Moreover, we will show that it satisfies also

for.

D_{j,k}(uv)=(D_{j+1,k+1}u)v+u(D_{j+2,k+2}v)(1.32).

We first calculate

g_{j}d_{k}(x, y)g_{j}^{-1}(uv)=g_{j}d_{k}(x, y)\{(g_{j+1}^{-1}u)(g_{j+2}^{-1}v)\}=g_{j}(\{d_{k+1}(x, y)(g_{j+1}^{-1}u)\}(g_{j+2}^{-1}v))+g_{j}((g_{j+1}^{-1}u)(d_{k+2}(x, y)g_{j+2}^{-1}v))=g_{j+1}(d_{k+1}(x, y)(g_{j+1}^{-1}u)g_{j+2}(g_{j+2}^{-1}v))+(g_{j+1}g_{j+1}^{-1}u)(g_{j+2}d_{k+2}(x, y)g_{j+2}^{-1}v)=g_{j+1}d_{k+1}(x, y)g_{j+1}^{-1}u)v+u(g_{j+2}d_{k+2}(x, y)g_{j+2}^{-1}v\}.
Similarly, we find\{d_{k}(g_{j-k}x, y)+d_{k}(x, g_{j-k}y)\}(uv)=\{d_{k+1}(g_{j-k}x, y)u\}v+u\{d_{k+2}(g_{j-k}x, y)v\}+\{d_{k+1}(x, g_{j-k})u\}v+u\{d_{k+2}(x, g_{j-k}y)v\},
and these prove the validity of Eq.(1.31).

if we choose k=3 in Eq.(1.31), we obtain

D_{j_{)}3}(uv)=(D_{j+1,1}u)v+v(D_{j+2,2}v)=0
which gives D_{j,3}=0,

provided that the condition (B) holds. On the other side, the choiceof k=1 or k=2 in Eq.(1.32) yields

0=u(D_{j+2,3}v)=(D_{j+1,3}u)v
for any j=1 , 2, 3 and for any u, v\in Acondition (C). \square.

Therefore, it also gives D_{j,3}=0 under the

We are now in position that we can construct a invariant
follow:

sub‐group of Tnig (A) as

Let A be a regular triality algebra satisfying either condition (B) or (C), and set

L_{0}=span <d_{j}(x, y) , \forall x, y\in A, \forall j=1, 2, 3>
Then L_{0} is a Lie algebra by \mathrm{E}\mathrm{q}.(1.29\mathrm{b}).

(1.33).

Moreover, it is a ideal of the larger Lie algebras\circ Lrt(A) by \mathrm{E}\mathrm{q}.(1.29\mathrm{a}) For any basis e_{1}, e_{2}\cdots, e_{N} of A with N=Dim A and for any$\alpha$_{j, $\mu,\ \nu$}\in F(j=1,2,3, $\mu$, \mathrm{v}=1,2, \cdots, N) We set.

,.

D_{j}=\displaystyle\sum_{k=1}^{3}\sum_{$\mu,\ nu$=1}^{N}$\alpha$_{j-k,$\mu,\ nu$}d_{k}(e_{$\mu$},e_{$\nu$})\inL_{0}(1.34)

15

for j=1 2, 3. Then, D=(D_{1}, D_{2}, D_{3}) is a member of s\mathrm{o}Lrt(A) by Eq.(1.29). Thereofore,

its exponential map$\xi$_{j}=exp D_{j}(j=1,2,3)(1.35)

satisfies $\xi$_{j}(xy)=($\xi$_{j+1}x)($\xi$_{j+2}y) , i.e., ($\xi$_{1}, $\xi$_{2}, $\xi$_{3})\in \mathrm{T}\mathrm{r}\mathrm{i}\mathrm{g}(A) , provided that the exponential

map is well‐defined.

Moreover, for any g=(g_{1}, g_{2}, g_{3})\in \mathrm{T}\mathrm{r}\mathrm{i}\mathrm{g}(A) We calculate.

g_{j}D_{k}g_{j}^{-1}=\displaystyle\sum_{l=1}^{3}\sum_{$\mu,\ nu$=1}^{N}$\alpha$_{k=l,$\mu,\ nu$}\{d_{l}(g_{j-l}e_{$\mu$},e_{$\nu$})+d_{j}(e_{$\mu$},g_{j-l}e_{$\nu$})\} inL_{0}(1.36)

by Eq.(1.30), so thatg_{j}$\xi$_{k}g_{j}^{-1}=g_{j}(expD_{k})g_{j}^{-1}=exp(g_{j}D_{k}g_{j}^{-1})\in exp L_{0}(1.37).

Thereofre, the group G_{0}=<$\xi$_{1}, $\xi$_{2}, $\xi$_{3}>\mathrm{i}\mathrm{s} a invariant sub‐group of Trig (A).

Remark 1.4
If a regular triality algebra satisfies \mathrm{E}\mathrm{q}.(1.23\mathrm{b}) as well asd_{3}(x, y)z+d_{3}(y, z)x+d_{3}(z, x)y=0(1.38),

then A is known as a pre‐normal triality algebra([K‐O.15]). Moreover, if we haveQ(x, y, z)=d_{1}(z, xy)+d_{2}(y, zx)+d_{3}(x, yz)=0(1.39)

in addition, A is called a normal triality algebra in [K‐O.15] also.

Further, suppose that A is involutive,with the involution map x\rightarrow\overline{x} as in Eq.(1.10)

and intoroduce the second bi‐linear product in the, same vector space of A by

x*y=\overline{xy}=\overline{y}\overline{x}(1.40).

Then, the resulting algebra A^{*} is also involutive. If A^{*} is unital, then A^{*} is the structurable
algebra of Allison ([A.78]), and both conditions (B) and (C) are automatically satisfied.
If we setD(x, y)=d_{0}(x, y)+d_{1}(x, y)+d_{2}(x, y),then for any element x, y of a structurable algebra A^{*} this D(x, y) satisfies the validityof a generalized structurable algebra([Ka 92
That is, it holds,

D(x, y) is a derivation, and D(x, yz)+D(y, zx)*D(z, xy)=0.
Finally, Eq.(l.l) is rewritten as a modified global triality relation of\overline{g}_{j}(x*y)\cdot=(g_{j+1}x)*(g_{j+2}y).

(1.41)

Reviewing these results, we can construct an invariant sub‐group of Trig (A) from any
regular triality algebra A satisfying condition (B) or (C), in particular from any struc‐turable algebra A^{*}.
However, for some cases, we can more directly construct some invariant sub‐group of
Trig (A) as it happenes for the case of A being a symmetric composition algebra. The casewill be explored in the next section. Ifwe further restrict ourselves to the automorphism,16group, then

we

can

also constructexplict automorphism of the Cayley algebra.

somewhich we will study in other paper, and also other case of Trig (A) will be given in future.

In the end of this section, we note that the normal triality algebra is useful concept

for a construction of Lie algebras (to see, our paper [J.Alg., 416 (2014) 58‐83]).

(2) Symmetric Composition AlgebraThis section is devoted for constructions of Trig (A) for symmetric composition alge‐bras. Let A be an algebra over a field F of characteristic not 2 with a symmetric bi‐linear

non‐degenerate form <>.

Suppose that it satisfies(xy)x=x(yx)=<x|x>y(2.1)

for any x, y\in A Then A is called a symmetric composition algebra, since it also satisfies.

the composition law<xy|xy>=<x|x><y|y> and <xy|z>=<x|yz>

(2.2)

.

Conversely, the validity of Eq.(2.2) implies that of Eq.(2.1) ([O,95]).
Linearizing Eq.(2.1), it gives(xy)z+(zy)x=x(yz)+z(yx)=2<x|z>y(2.3)

for any x, y, z\in A Moreover if we replace z by yz in the first relation of Eq.(2.3) and.

note

Eq.(2.2), it yields( xy )( yz ) =2<x|yz>y-<y|y>zx
Also, for any g=(g_{1}, g_{2}, g_{3})\in \mathrm{T}\dot{\mathrm{n}}\mathrm{g}(A)

,

(2.4)

.

have

we

<g_{j}x|g_{j}y>=<x|y>(j=1,2,3)(2.5)

by the following reason. Applying g_{j} to both sides of Eq.(2.1), it gives<x|x>g_{j}y=g_{j} { x (yx)} =(g_{j+1}x)g_{j+2}(yx)=(g_{j+1}x)\{(g_{j}y)(g_{j+1}x)\}=<g_{j+1}x|g_{j+1}x>g_{j}yso that we havewe<x|z>=<g_{j+1}x|g_{j+1}x> Linearizing the relation, and letting j\rightarrow j-1,
Similarly, any t_{j}\in s\mathrm{o}Lrt(A) satisfies.

obtain then Eq.(2.5).

<t_{j}x|y>=-<x|t_{j}y>. (j=1,2,3)(2.6)

Any symmetric composition algebra is known (see [0‐O,8I],[E,97], [K‐M‐R‐T,98]) to
algebra or 8‐dimensional pseudo‐octonion algebra where theisthepara‐Hurwitz algebra
conjugate algebra of Hurwitz algebra with the para‐unit e
ex=xe
satisfying=\overline{x}=2<e|x>e-x Moreover, it is a normal triality algebra
conditions([O,05]). Further,(B)and (C) of section 1 are automatically satisfied by thisalgebra. Since <\circ|\circ> is assumed to be non‐degenerate, there exists x\in A satisfying<x|x>\neq O. Now Eq.(2.1) implies <x|x>y\in AA for any element y\in A so that we
be either a para‐Hurwitz

.

,

17

have AA=A If b\in A satisfies bA=0
.

b=0

.

.

then/Eq.(2.1) also leads to <x|x>b=0 and here

Therefore both conditions (B) and (C) are automatically.satisfied.

Especially, any

symmetric composition algebra is a regular triality algebra satisfying both conditions (B)and (C) (see Remark 1.4). Thus, we can construct some invariant sub‐group of Trig (A) as
has been demonstrated in the previous sections. However, for special case of Dim A=1,
we can construct entire Txig
(A) as follows.

Example 2.1. Case of Dim A=1
We write A=Fe, where e\in A satisfies

with

ee=e

g=(g_{1}, g_{2}, g_{3})\in \mathrm{h}\mathrm{i}\mathrm{g}(A) we can write<e|e>=1.

Then for any,g_{j}(e)=$\alpha$_{j}e, (j=1,2,3)
for some $\alpha$_{j}=$\alpha$_{j+1}$\alpha$_{j+2} while Eq.(2.5) yields ($\alpha$_{j})^{2}=1.

Therefore, there gives$\alpha$_{1}^{2}=$\alpha$_{2}^{2}=$\alpha$_{3}^{2}=1 and $\alpha$_{1}$\alpha$_{2}$\alpha$_{3}=1
sothat Trig(Fe) is isomorphic to the Kleins 4‐group K_{4}That is,.

Trig (Fe)\cong< ( Id Id, Id), (Id, -Id, -Id) (-Id, Id, -Id) (-Id, -Id, Id)>_{span}.
,,,

Note that we haveAuto(Fe) =<Id>
being trivial.
For the case of Dim A=2 , we will discuss in future.

This paper is an announcement of our recent works, and the details will be consideredin future.

3 Other Examples of Triality Groups

Here, in this section, we will give some examples of the triality group for algebras
other than the symmetric composition algebra.

Example 3.1 (Matrix algebra)
Let M(n, F) be a set consisting all n\times n matrices over the field F For any x, y\inM(n, F) , the matrix product which we designate as x*y is associative and we write.

x*(y*z)=(x*y)*z :=x*y*z
Moreover, for the transpose matrix t_{X} of any x\in M(n, F)\overline{x}=^{t}x

we

set

(3.2)

.

Then x\rightarrow\overline{x} is a involution map of the resulting algebra A^{*}
associative algebra.

,

(3.1)

.

.

Then, A^{*} is a unital involutive

Especially, it is structurable ([A‐F,93]).

Note that the n\times n unit

matrix e is, here, the unit element of A^{*} We introducea subset of A^{*} by.

A_{0}^{*}=\{x|\overline{x}*x=x*\overline{x}=e, x\in A^{*}\}

.

(3.3)

18

For any three a_{j}\in A_{0}^{*},(j=1,2,3) we introduce $\sigma$_{j}(a)\in \mathrm{E}\mathrm{n}\mathrm{d}A^{*} by,$\sigma$_{j}(a)x :=a_{j}*x*\overline{a}_{j+1}, (j=1,2,3)(3.4)

where the indices over j are defined modulo 3, i.e., a_{j\pm 3}=a_{j} It is easy to see the validity.

of$\sigma$_{j}(a)$\sigma$_{j}(\overline{a})=1, (3.5a)\overline{$\sigma$_{j}(a)}x=a_{\mathrm{j}+1}*x*\overline{a}_{j}, (3.5b)\overline{$\sigma$_{j}(a)}(x*y)=($\sigma$_{j+1}(a)x)*($\sigma$_{j+2}(a)y)(3.5c).

Introducing the conjugate algebra A of A^{*} with the bi‐linear product xy by
xy=\overline{x*y}=\overline{y}*\overline{x}

(3.6)

.

then, these are rewritten as$\sigma$_{j}(a)x=a_{j+1} (ajx) =(x\overline{a}_{j+1})\overline{a}_{j}(3.7a)$\sigma$_{\mathrm{j}}(a)(xy)=\{($\sigma$_{j+1}(a)x)\}\{($\sigma$_{j+2}(a)y)\}. (3.7b)
We note that A is no longer associative but para‐associative with the para‐associative lawof\overline{z}(xy)=(yz)\overline{x}

(3.8)

.

Fkom \mathrm{E}\mathrm{q}\mathrm{s}.(3.5\mathrm{a}) and (3.7b), we find$\sigma$(a)=($\sigma$_{1}(a), $\sigma$_{2}(a), $\sigma$_{3}(a) \in \mathrm{T}\mathrm{r}\mathrm{i}\mathrm{g}(A)

(3.9)

.

Furthermore, \mathrm{E}\mathrm{q}.(3.7\mathrm{a}) gives$\sigma$_{j}(a)=L(a_{j+1})L(a_{j})=R(\overline{a}_{j})R(\overline{a}_{j+1})(3.10),

which has the same structures as the relations for the symmetric composition algebra.
For the corresponding to local triality case, let

p=(p_{1},p_{2},p_{3})\in(A^{*})^{3}, \overline{p}_{j}=-p_{j}(j=1,2,3)
and define(3.11)

d_{j}(p)\in End A^{*} byd_{j}(p)x=p_{j}*x-x*p_{j+1}(3.12).

We then have\overline{d_{j}(p)}x=x*\overline{p}_{j}-\overline{p}_{j+1}*x=p_{j+1}*x-x*p_{j} (3.13a)\overline{d_{j}(p)}(x*y)=(d_{j+1}(p)x)*y+x*(d_{:i+2}(p)y) , (3.13b)
and henced_{j}(p)(xy)=(d_{j+1}(p)x)y+x(d_{J+2}(p)y)\mathrm{T}_{$\eta$}\mathrm{h}ese imply that d(p)=(d_{1}(p), d_{2}(p), d_{3}(p))\in s\mathrm{o}Lrt(A)However,(3.14).

This implies á version of matrix algebras for well‐known
we

.



the principle of triality

will not go into details

In final of this section, we give some more simple examples for the triality group.

19

Example 3.2

Let \mathrm{C} be the complex number with usual product x*y.

If we define

for a=( $\alpha$, $\beta,\ \gamma$)\in \mathrm{C}^{3} , and | $\alpha$|=| $\beta$|=| $\gamma$|=1,$\sigma$_{1}(a)x= $\alpha$*x*$\beta$^{-1}$\sigma$_{2}(a)x= $\beta$*x*$\gamma$^{-1}$\sigma$_{3}(a)x= $\gamma$*x*$\alpha$^{-1}
where $\sigma$_{j\pm 3}=$\sigma$_{j}(j=0,1,2) then we have,$\sigma$_{j}(a)(xy)=($\sigma$_{\mathrm{j}+1}(a)x)($\sigma$_{j+2}(a)y)
with respect to new product xy defined by xy=\overline{x*y},where \overline{x} denotes the conjugation of x.
If we introduce for a=( $\alpha.\ \beta$, $\gamma$)\in( Im \mathrm{C})^{3}

d_{1}(a)x= $\alpha$*x-x* $\beta$, d_{2}(a)x= $\beta$*x-x* $\gamma$, d_{3}(a)x= $\gamma$*x-x* $\alpha$,then we getd_{j}(a)(xy)=(d_{j+1}(a)x)y+x(d_{j+2}(a)y).

Example 3.3 Let \mathrm{H} be the quaternion number satisfying i^{2}=j^{2}=k^{2}=-1, i*j=
-j*i=k with basis \{1, i, j, k\} If we define for a=(i,j, k)\in ( Im \mathrm{H})^{3},.

$\sigma$_{1}(a)x=i*x*j^{-1}$\sigma$_{2}(a)x=j*x*k^{-1}$\sigma$_{3}(a)x=k*x*i^{-1}
where the product x*y is usual product of \mathrm{H},then we have$\sigma$_{j}(xy)=($\sigma$_{j+1}(a)x)($\sigma$_{j+2}(a)y)
w.r. \mathrm{t}.

newproduct xy=\overline{x*y}where \overline{x} denotes the involutive conjugation of x\in \mathrm{H}.

If we introduce d_{1}(a)x=i*x-x*j, d_{2}(a)x=j*x-x*k, d_{3}(a)x=k*x-x*i then,weobtaind_{j}(a)(xy)=(d_{j+1}(a)x)y+x(d_{j+2}(a)yw.r. \mathrm{t} newproduct xy defined by xy=\overline{x*y}.

Finally, hereby we would like to note final words of Prof.S.Okubo;
Jisei no ku
To be or not to be? The quantum dream of the Sohrodinger Cat.

Farewell! Farewell!! forever.

Departure time now to the black hole.

Never to return, farewell, sayonara.

This paper is a cowork with Prof. S.Okubo in the end of his life.
This note is an announcement and the details will be worked in future.

20\mathrm{R}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{s}\backslash[A‐78]: Allison,B.N.; A class of non‐associative algebras with involution containing aMath,Ann 237, (1978) 133‐156[A‐F.93]: Allison,B.N.,Faulkner;J.R.; Non‐associative coefficient algebras for Steinberg
unitary Lie algebras J.Algebras 161 (\mathrm{I}993)133-158[E‐97]: Elduque,A.; Symmetric Composition Algebra J.Algebra 196(1997) 282‐300[E,00]: Elduque,A.; On triality and atuomorphism and derivation of composition alge‐bra Linear algebra and its applications 314(2000) 49‐74[J.58]: Jacobson,N. Composition algebras and thier automorphisms Rend. Circ. Mat.
Palermo 7 (1958),55-80[Ka.92]: Kamiya,N. On a generalization of structurable algebras Algebras, Groups andgeometries 9 (1992) 35‐47.
[K‐M‐P‐T.98]: Knus,M.A.,Merkurjev,A.S.,Post,M.,Tignal,J.P; The Book of Involution
American Math.Soc.Coll.Pub. 44 Providence (1998)[K‐O.15]: Kamiya,N.Okubo,S.; Algebras satisfying triality and S_{4} Symmetry to appear(2015) Arxiv. 1503.00614[O.05]: Okubo,S.; Symmetric triality relations and structurable algebra Linear Algebras
and its Applications, 396 (2005) 189‐222[O‐0.81]: Okubo,S.,Osborn,J.M.; Algebras with non‐degenerate associative symmetricbi‐linear form permitting compositions Comm.Algebra 9 (1981) (I) 1233‐1261, (II) 2015‐
class of Jordan algebra

\cdot



2073

[S.66]:Schafer,R.D.; An Introduction to Non‐associate algebra Academic press.

and London (1966)

N.Y.