2:282 RIGHT ALGEBRA

The actions of the left stabilizers of 2:282 ILC on its left XLC were abstracted into a unital algebra by way of a set of commutators and associators. In a similar way, the corresponding right algebra over the same ground field F12697 can be derived via the actions of the ILC right-stabilizers on the right XLC.

Let V be the 2:282's right XLC: (9851, 9816, 1553). The right-stabilizers S_i, 1 ≤ i ≤ 4, likewise, act on V (considered as a column-vector) via matrix-vector multiplications:

$S$₁ V = $\left(\begin{array}{ccc}6442& 8313& 618\\ 11024& 11771& 6929\\ 5220& 6653& 176\end{array}\right)$ $\left(\begin{array}{c}9851\\ 9816\\ 1553\end{array}\right)$ $=$ $\left(\begin{array}{c}4804\\ 7797\\ 11238\end{array}\right)$

$S$₂ V = $\left(\begin{array}{ccc}8566& 6209& 11692\\ 5399& 3926& 4546\\ 3585& 3137& 10438\end{array}\right)$ $\left(\begin{array}{c}9851\\ 9816\\ 1553\end{array}\right)$ $=$ $\left(\begin{array}{c}2314\\ 443\\ 4190\end{array}\right)$

$S$₃ V = $\left(\begin{array}{ccc}12657& 4518& 10592\\ 12194& 12284& 2208\\ 5931& 8638& 9879\end{array}\right)$ $\left(\begin{array}{c}9851\\ 9816\\ 1553\end{array}\right)$ $=$ $\left(\begin{array}{c}4395\\ 6643\\ 11537\end{array}\right)$

$S$₄ V = $\left(\begin{array}{ccc}11485& 1339& 6251\\ 9768& 4423& 4791\\ 3114& 3474& 7660\end{array}\right)$ $\left(\begin{array}{c}9851\\ 9816\\ 1553\end{array}\right)$ $=$ $\left(\begin{array}{c}5292\\ 11808\\ 8292\end{array}\right)$

The column-vectors on the right-hand-side are the S_i actions on V.

The commutators, [S_i, S_j] = S_iS_j − S_jS_i, as linear operators, are determined as before via matrix-matrix multiplications:

[S₁, S₂] = $\left(\begin{array}{ccc}7994& 5455& 1752\\ 722& 11919& 10223\\ 6307& 7208& 5481\end{array}\right)$

[S₁, S₃] = $\left(\begin{array}{ccc}893& 12424& 366\\ 8227& 12573& 9549\\ 9396& 2250& 11928\end{array}\right)$

[S₁, S₄] = $\left(\begin{array}{ccc}5122& 616& 9680\\ 10926& 8631& 7397\\ 9815& 936& 11641\end{array}\right)$

[S₂, S₃] = $\left(\begin{array}{ccc}9932& 173& 4670\\ 6519& 4003& 7477\\ 6372& 9489& 11459\end{array}\right)$

[S₂, S₄] = $\left(\begin{array}{ccc}10931& 6693& 7410\\ 5812& 10342& 12395\\ 3048& 3319& 4121\end{array}\right)$

[S₃, S₄] = $\left(\begin{array}{ccc}7566& 10614& 9448\\ 10649& 11538& 2164\\ 4853& 2452& 6290\end{array}\right)$

The corresponding actions of the commutators on (the column-vector) V = (9851, 9816, 1553) are determined via the usual matrix-vector multiplications:

[S₁, S₂] V = $\left(\begin{array}{c}8829\\ 1220\\ 2186\end{array}\right)$

[S₁, S₃] V = $\left(\begin{array}{c}6951\\ 455\\ 4044\end{array}\right)$

[S₁, S₄] V = $\left(\begin{array}{c}1620\\ 3825\\ 5700\end{array}\right)$

[S₂, S₃] V = $\left(\begin{array}{c}9040\\ 199\\ 2766\end{array}\right)$

[S₂, S₄] V = $\left(\begin{array}{c}6482\\ 8579\\ 9567\end{array}\right)$

[S₃, S₄] V = $\left(\begin{array}{c}4427\\ 9137\\ 2395\end{array}\right)$

The actions on V presented plainly as column-vectors are not particularly insightful since the relationships that may exist between commutators and stabilizers remain hidden. To uncover these correlations, the actions [S_i, S_j] V will be recast as before in terms of special bases formed with V and complementary S_i actions: {V, S_kV, S_lV} where k and l are set-complement of i and j: {i, j} ∪ {k, l} = {1, 2, 3, 4}. The vectors on the right hand side take the following forms in their respective bases:

[S₁, S₂] V = (8829, 1220, 2186) = 2518 V + 5986 S₃V + 7961 S₄V
[S₁, S₃] V = (6951, 455, 4044) = 1300 V + 1165 S₂V + 966 S₄V
[S₁, S₄] V = (1620, 3825, 5700) = 3843 V + 8281 S₂V + 5750 S₃V
[S₂, S₃] V = (9040, 199, 2766) = 4274 V + 8314 S₁V + 5137 S₄V
[S₂, S₄] V = (6482, 8579, 9567) = 12072 V + 5187 S₁V + 4726 S₃V
[S₃, S₄] V = (4427, 9137, 2395) = 513 V + 8054 S₁V + 324 S₂V

for the right XLC V = (9851, 9816, 1553).

The above actions on the right XLC, V, are then abstracted into a nonassociative algebra with formal generators X_i corresponding to S_i and their actions. The commutators on X_i's mirror those of S_i's as before:

[X₁, X₂] = 2518   + 5986 X₃ + 7961 X₄
[X₁, X₃] = 1300   + 1165 X₂ +   966 X₄
[X₁, X₄] = 3843   + 8281 X₂ + 5750 X₃
[X₂, X₃] = 4274   + 8314 X₁ + 5137 X₄
[X₂, X₄] = 12072 + 5187 X₁ + 4726 X₃
[X₃, X₄] =   513   + 8054 X₁ +   324 X₂

The resulting unital (right) algebra is thus the formalization of the actions of the right stabilizers of 2:282's ILC on its right XLC. With the same ordering, X₁ > X₂ > X₃ > X₄, in conjunction with the degrees of the monomials, the polynomials can be uniquely presented in descending monomial order. As the commutators are specified free from additional constraints, the algebra is, subsequently, nonassociative. The following is the list of the corresponding associators:

[X₃, X₂, X₁] = + 2717 X₁ − 6277 X₂ − 2565 X₃ + 206
[X₄, X₂, X₁] = − 2030 X₁ − 1531 X₂ + 2565 X₄ + 6200
[X₄, X₃, X₁] = + 6017 X₁ + 1531 X₃ − 6277 X₄ + 1378
[X₄, X₃, X₂] = − 6017 X₂ − 2030 X₃ − 2717 X₄ + 5777

The associators, thus, take the same general form:

$[X$_k, X_j, X_i] =

with constants a_ijk, b_ijk, c_ijk, d_ijk as given in the above four equations.