LINEAR BRACKET OPERATORS

The left and right (co)-bif-2 algebras on chapter two were defined over the finite field F12697 with basis 2:{0, 1, 286}. The algebras were then trasported via the bijective linear map Q to their corresponding left/right ALM LC-algebras with standard basis {A, L, M} over the same ground field. The nonassociativity of the algebras led to the introduction of a number of bracket operators to formally manipulate parenthesizations of product expressions. Their actions within the LC-algebras were facilitated via a suitably constructed bracket bases (BK).

There are five bracketings, (0X6A, 0X6C, 0X72, 0X74, 0X78), that can be applied to any products of four elements in the LC algebra. In particular, they will be applied to the four ALM-words of the tetron in the order of their appearances. Their actions on both cases left-to-right (M ALL ALM ALM) and right-to-left (MLA MLA LLA M) will be considered. The resulting pair of sets of five LCs each form the pentagonal prism with vertices denoted as V6A, V6C, V72, V74, V78. The vertical associators are then obtained with pairs of corresponding LCs from the top and bottom faces. The three associators formed with V72, V6C, V74, also denoted as B1, B2, B3, resp. will be taken as the basis: BK = {B1, B2, B3}:

Left-to-right:

B1 = (M((AL.L)(AL.M)))(AL.M) − (M((A.LL)(AL.M)))(AL.M)     -V72-
B2 = (M(AL.L))((AL.M)(AL.M)) − (M(A.LL))((AL.M)(AL.M))     -V6C-
B3 = M(((AL.L)(AL.M))(AL.M)) − M(((A.LL)(AL.M))(AL.M))     -V74-

Right-to-left:

B1 = ((M.LA)((M.LA)(LL.A)))M − ((M.LA)((M.LA)(L.LA)))M     -V72-
B2 = ((M.LA)(M.LA))((LL.A)M) − ((M.LA)(M.LA))((L.LA)M)     -V6C-
B3 = (M.LA)(((M.LA)(LL.A))M) − (M.LA)(((M.LA)(L.LA))M)     -V74-

The BKs will be realized in both (left and right) LC-algebras.

(Left) LC-Algebra:

Left-to-Right:

B1 = (9982, 9610, 11680)
B2 = (11113, 6155, 9948)
B3 = (2971, 9664, 10709)

Right-to-Left:

B1 = (12297, 7235, 3954)
B2 = (4508, 2291, 1221)
B3 = (3993, 3472, 10867)

(Right) LC-Algebra:

Left-to-Right:

B1 = (179, 3603, 5449)
B2 = (372, 1026, 11796)
B3 = (8361, 8245, 10892)

Right-to-Left:

B1 = (2710, 8348, 7684)
B2 = (95, 12554, 5081)
B3 = (8677, 6320, 3447)

In fully expanded form the last one, for instance, takes the following form:

B3 = 8677 A + 6320 L + 3447 M.

Among the bracket operators, the complementation operators Ci's will be picked to act on the BK. Since, Bi's are derived from the products of four ALM-words there are four corresponding complement operators: Ci, 1 ≤ i ≤ 4. The following chart gives their actions on the Bi's:

C1 B1 = B2
C2 B1 = V6A
C3 B1 = B3
C4 B1 = V78

C1 B2 = B3
C2 B2 = B2
C3 B2 = B2
C4 B2 = B1

C1 B3 = B1
C2 B3 = V6A
C3 B3 = B2
C4 B3 = V78

Next, is to apply Ci's to arbitrary LCs: a A + l L + m M or equivalently (a, l, m) with coefficients a, l, m in the finite field. This can be achieved via expressing the standard basis letters A, L, and M in terms of the BK. Then the actions extend linearly to all LCs. The basis letters in component form are:

A = 1 A + 0 L + 0 M = (1, 0, 0)
L = 0 A + 1 L + 0 M = (0, 1, 0)
M = 0 A + 0 L + 1 M = (0, 0, 1)

Their expansions in the left-LC-algebra/left-to-right BK is:

A = (1, 0, 0) = 1521 B1 + 290 B2 + 8267 B3
L = (0, 1, 0) = 8661 B1 + 4757 B2 + 11639 B3
M = (0, 0, 1) = 1948 B1 + 5611 B2 + 7135 B3

It follows that the Ci action on an LC is:

Ci (a, l, m) = Ci (a A + l L + m M) = a Ci A + l Ci L + m Ci M.

Substituting for A, L, and M in terms of their BK expansions gives

Ci (a, l, m) =

a Ci (1521 B1 + 290 B2 + 8267 B3) +
l Ci (8661 B1 + 4757 B2 + 11639 B3) +
m Ci (1948 B1 + 5611 B2 + 7135 B3) =

a (1521 Ci B1 + 290 Ci B2 + 8267 Ci B3) +
l (8661 Ci B1 + 4757 Ci B2 + 11639 Ci B3) +
m (1948 Ci B1 + 5611 Ci B2 + 7135 Ci B3)

Substituting again for the actions of C1, C2, C3, C4 on B1, B2, B3 from the above chart fully determine their action:

C1 (a, l, m) =

a (1521 C1 B1 + 290 C1 B2 + 8267 C1 B3) +
l (8661 C1 B1 + 4757 C1 B2 + 11639 C1 B3) +
m (1948 C1 B1 + 5611 C1 B2 + 7135 C1 B3) =

a (1521 B2 + 290 B3 + 8267 B1) +
l (8661 B2 + 4757 B3 + 11639 B1) +
m (1948 B2 + 5611 B3 + 7135 B1) =

a (4751, 1330, 1501) + l (10667, 5177, 9573) + m (2956, 3339, 2769) =

(4751 a + 10667 l + 2956 m, 1330 a + 5177 l + 3339 m, 1501 a + 9573 l + 2769 m).

The final expression can be written in matrix form where the following C1 matrix

C1 = ( 4751 10667 2956 1330 5177 3339 1501 9573 2769 )

acts on (a, l, m) via matrix-vector multiplication. The LC, (a, l, m), is treated as a column vector. The C1 bracket operator which originally acted purely on bracketings is now represented in the algebra by a matrix that can act linearly on arbitrary vectors via matrix multiplication.

All remaining Ci's are obtained similarly. The following is the list of corresponding matrices of all complementation operators:

(Left) LC-Algebra:

Left-to-Right:

C1 = ( 4751 10667 2956 1330 5177 3339 1501 9573 2769 )

C2 = ( 12364 3959 3043 9687 4573 9343 3855 4867 3139 )

C3 = ( 4867 1810 8939 9694 2904 5385 2486 810 4927 )

C4 = ( 344 3550 3271 324 6123 7076 11503 5502 5645 )

Right-to-left:

C1 = ( 5596 11615 569 12670 8700 9630 10253 12311 11098 )

C2 = ( 3400 2532 1137 5704 7081 4579 803 11586 7128 )

C3 = ( 237 11758 433 10807 1902 4139 11193 5690 10559 )

C4 = ( 1290 3048 11865 5437 4425 12029 1941 6632 6983 )

(Right) LC-Algebra:

Left-to-Right:

C1 = ( 8961 10909 4036 1870 5475 9675 12040 7966 10958 )

C2 = ( 8590 1897 3 8555 3284 4424 712 4822 7477 )

C3 = ( 12228 9562 6636 9873 6477 4757 494 1772 6690 )

C4 = ( 1016 10009 2657 12486 9304 9024 179 3671 9885 )

Right-to-left:

C1 = ( 11800 2338 7656 1074 1261 2678 5397 7498 12333 )

C2 = ( 9545 6280 9924 10441 3210 4738 7009 11528 8036 )

C3 = ( 11845 11926 7496 804 10105 9123 745 9569 3445 )

C4 = ( 7603 7952 1654 6181 7498 9563 1569 7010 92 )