The bracket complement operators led to the derivations of a special set of operators, the stabilizers Si's, preserving the 2:282 ILC: Si ILC = ILC. While the stabilizers act trivially on the ILC, their actions on the XLCs are nontrivial: Si XLC ≠ XLC. These nontrivial actions will be brought together in the form of algebras that, additionally, are unital as they contain the identity operator (Id) which (trivially) preserves the ILC (and all other vectors). They are also noncommutative in view of the actions of the stabilizers, as matrix multiplication in general is not commutative: (Si Sj ≠ Sj Si).
Associated with the verse 2:282 are its pair of left and right XLCs: (8735, 7287, 595) and (9851, 9816, 1553), resp., over the same ground field F12697, as before. The left/right stabilizers acting on the matching left/right XLCs would generate a pair of corresponding left/right algebras. The noncommutativity of the operators will be made explicit via the commutators [Si, Sj] = SiSj − SjSi. The commutators explicitly indicate the extend to which the product fails to be commutative. It is zero when it is commutative. Since, any element commutes with itself, [Si, Si] = 0. With [Si, Sj] = − [Sj, Si], it suffices to consider only the six [Si, Sj], for 1 ≤ i < j ≤ 4.
Let V be the left XLC, (8735, 7287, 595), considered as a column
vector. Then the actions of left Si's are:
,
where S1 is represented by its matrix that was determined
previously. The vector (5725, 3471, 8170) on the right-hand-side is the
result of the action of S1 on V. The remaining left
Si's act similarly:
If P and Q are two arbitrary linear operators, so are P ± Q, PQ and QP.
The latter two indicate operator compositions in which they are applied in
successions from right to left. Consequently, the commutators
[Si, Sj] themselves are linear operators. Their
compositions can readily be determined via matrix multiplications:
[S1, S2] =
S1 S2 − S2 S1,
where,
S1 S2 =
=
As linear operators, commutators act on V through
matrix-vector multiplications:
[S1, S2] V =
=
[S1, S3] V =
=
The column-vectors on the right-hand-sides are the actions of the commutators on V. At first glance they appear detached and unrevealing, offering no clues at possible links that may exist between commutators and stabilizers. This can be rectified if the actions are expressed in terms of appropriately chosen bases that would make such correlations explicit. The bases would consist of V and a complementary pair of stabilizers actions on it, SkV and SlV, obtained above. V can be seen as the trivial action of the identity operator: V = Id V.
The commutators actions on the XLC, [Si, Sj] V,
will be expressed in terms of the distinguished basis
{V, SkV, SlV} where {k, l} is the set-complement
of {i, j}: {i, j} ∪ {k, l} = {1, 2, 3, 4}:
[S1, S2] V =
aV + bS3V + cS4V
[S1, S3] V =
dV + eS2V + fS4V
[S1, S4] V =
gV + hS2V + iS3V
[S2, S3] V =
jV + kS1V + lS4V
[S2, S4] V =
mV + nS1V + oS3V
[S3, S4] V =
pV + qS1V + rS2V
where the 18 scalar coefficients, a - r, come from
F12697. Since, all vectors can be uniquely expressed in
terms of the basis vectors, the coefficients are fully determined.
The following is the list with their actual values
[S1, S2] V = (2608, 11969, 2451) =
70 V + 1360 S3V + 10336 S4V
[S1, S3] V = (4588, 469, 6481) =
10913 V + 5059 S2V + 6662 S4V
[S1, S4] V = (7763, 4478, 5900) =
665 V + 2227 S2V + 9475 S3V
[S2, S3] V = (6644, 4270, 10977) =
3880 V + 11318 S1V + 12012 S4V
[S2, S4] V = (12614, 10164, 10699) =
2647 V + 1587 S1V + 5788 S3V
[S3, S4] V = (6503, 8243, 11672) =
7822 V + 6227 S1V + 9759 S2V
for the left XLC V = (8735, 7287, 595). The commutators actions appear
as row vectors in the middle. They are expressed as linear combinations
of the basis vectors V, SkV and SlV with indices
{k, l}, complement of {i, j}.
The actions of the stabilizers and commutators on the left XLC V can be
abstracted into a not necessarily associative algebra whose generators
Xi's formally correspond to the stabilizers Si's.
The noncommutativity of the Xi's is specified via
the commutators that mirror those of the stabilizers:
[X1, X2] =
70 + 1360 X3 + 10336 X4
[X1, X3] =
10913 + 5059 X2 + 6662 X4
[X1, X4] =
665 + 2227 X2 + 9475 X3
[X2, X3] =
3880 + 11318 X1 + 12012 X4
[X2, X4] =
2647 + 1587 X1 + 5788 X3
[X3, X4] =
7822 + 6227 X1 + 9759 X2
where V has been "factored out". The resulting (left) algebra, basically
is the encoding and formalizing the actions of the left stabilizers of
2:282's ILC on its left XLC. It is a polynomial algebra of four
noncommuting indeterminates Xi's obeying the above commutation
relations. The generators Xi's can be thought of as independent
abstract symbols that formally represent Si's and their actions
on V but, otherwise, with no further relationships.
Noncommutative polynomials are finite linear combinations of monomials, the ordered product of not necessarily distinct Xi's: XiXj..Xk. The degree of a monomial is the number of the generators (with repetition) Xi's it contains. The degree of a polynomial is the maximum degree of its monomials. Since, Xi's do not commute the order of their appearances in the monomial is significant. Repeated adjacent generators can be denoted in exponential notation: X2X4X4X4X1 = X2X43X1, as an example of a fifth degree monomial.
A polynomial in one indeterminate, X, is a linear combinations of various powers of X or terms. The leading term is the one with the highest power which is also the degree of the polynomial. The powers of X define an ordering on terms: 1 < X < X2 < X3 < ... where 1 can be thought of X0. A given polynomial can then be represented with terms arranged uniquely in descending order, e.g. a cubic polynomial: a X3 + b X2 + c X + d, with a X3 as the leading term. Similarly, in polynomials with several noncommuting indeterminates there is a need to define the leading term. An ordering on terms (monomials) allow for systematically simplifying expressions and displaying the polynomials with monomials arranged in a unique descending order.
Noncommutative monomials are treated like words formed with the generators. To express the monomials uniquely in descending order, a specific ordering on the indeterminates is chosen: X1 > X2 > X3 > X4, in conjunction with the degree of the monomial. Accordingly, a monomial is considered in normal form when the generators appear in non-ascending order, e.g. XiXjXkXlXm, with i ≤ j ≤ k ≤ l ≤ m. A general monomial in normal form is, therefore, of the form X1n1 X2n2 X3n3 X4n4 of total degree n = n1 + n2 + n3 + n4, with ni ≥ 0. A generator with zero exponent can simply be omitted since Xi0 = 1. Monomials not in normal form can be algebraically manipulated, (moving generators into order), via repeated applications of the commutation relations until they are reduced to normal form. A reduced monomial can not be further simplified as it is already in normal form.
As an example, monomial X1X2 is normal. However,
X2X1 is not. It can be reduced with the help of
the commutator [X1, X2] =
X1X2 − X2X1 =
70 + 1360 X3 + 10336 X4 giving
X2X1 in terms of X1X2
and lower degree terms:
X2X1 = X1X2
− (70 + 1360 X3 + 10336 X4) =
X1X2 + 11337 X3 +
2361 X4 + 12627.
All terms on the right-hand-side are in normal form. Similarly,
the expression X2X1X3 can be
reduced via the same commutator:
X2X1X3 =
(X2X1)X3 =
(X1X2 + 11337 X3 +
2361 X4 + 12627)X3
=
X1X2X3 +
11337 X32 + 2361 X4X3
+ 12627 X3.
All terms on the right-hand-side are normal except the term with
X4X3. Similar use of the corresponding
commutator [X3, X4] =
7822 + 6227 X1 + 9759 X2 gives
the final reduced expression:
X2X1X3 =
X1X2X3 +
11337 X32 + 2361 X3X4 +
1179 X1 + 4056 X2 + 6393
In contrast, reducing the monomial
X3X2X1 into normal form is not
uniquely determined since it can be achieved with either
X3X2 or X2X1
leading to different expressions. The reduction sequence,
therefor, is path dependent and bracketing the monomial to pick
a specific path, (X3X2)X1 or
X3(X2X1), can be interpreted as
nonassociativity. This suggests that given a set of commutators with
no additional constraints, the product is not necessarily going to be
associative, in general. The difference between the two bracketings
can be regarded as a measure of the deviation from associativity and
will be made explicit via the associator
[X3, X2, X1] =
(X3X2)X1 −
X3(X2X1). The following is the
list of all such associators:
[X3, X2, X1] =
+ 5186 X1 + 2201 X2 −
1075 X3 − 2918
[X4, X2, X1] =
+ 991 X1 − 1665 X2 +
1075 X4 − 6150
[X4, X3, X1] =
− 2488 X1 + 1665 X3 +
2201 X4 − 4243
[X4, X3, X2] =
+ 2488 X2 + 991 X3 −
5186 X4 − 1118
The associators, thus, take the following general form:
The constants aijk, bijk, cijk,
dijk are those listed in the above four equations. The left
algebra so derived represents the actions of the four left stabilizers of
the ILC of 2:282 on its left XLC. It is unital, noncommutative and
nonassociative, defined by commutators and associators.