On the Octonion-like Associative Division Algebra
Juhi Khalid, Martin Bouchard School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward, Ottawa K1N6N5 Canada
ABSTRACT
Using elementary linear algebra, this paper clarifies and proves some concepts about a recently introduced octonion-like associative division algebra over .
The octonion-like algebra is the even subalgebra of Clifford algebra Cl40, () ,
which is isomorphic to Cl03, () and to the split-biquaternion algebra. For two seminorms described in the paper (which differ from the norm used in the original papers on the octonion-like algebra), it is shown that the octonion-like algebra is a seminormed algebra over with no zero divisors when using one of the two seminorms. Moreover, additional results related to the computation of inverse numbers in the octonion-like algebra are introduced in the paper, confirming that the octonion-like algebra is a division algebra over as long as the two seminorms are non-zero. Additional results on normalization of octonion-like numbers and some involutions are also presented. The elementary linear algebra descriptions used in the paper also allow straightforward software implementations of the octonion-like algebra.
KEYWORDS: associative octonion-like algebra, octonion-like seminormed algebra, octonion-like division algebra, hypercomplex algebra.
I. INTRODUCTION
In [1], [2], an eight-dimensional octonion-like associative division algebra over was recently introduced and presented as a normed algebra. There has been some controversy with this octonion-like algebra, as it was noted that a new normed division algebra over would contradict Hurwitz’s theorem [3]. It was pointed out that the octonion-like algebra, which is the even subalgebra of Clifford algebra Cl40, () , is isomorphic to Cl03, () [4] which,
1 in turn, is isomorphic to the split-biquaternion algebra and to the sum of two quaternion algebras [5]. The octonion-like algebra presented in [1], [2] was also found to be the same as the 1d-up approach to conformal geometric algebra presented in [6].
Using elementary linear algebra and two scalar seminorms defined in this paper, we explain that the associative octonion-like algebra over is a seminormed algebra and it is also a division algebra as long as the two seminorms are non-zero. Therefore, as an associative seminormed division algebra over , the octonion-like algebra does not contradict Hurwitz’s theorem. It should be noted that the two scalar seminorms used in this paper differ from the norm used in the original papers on the octonion-like algebra [1], [2], where a multi-dimensional non-scalar norm was first defined and subsequently constrained to become a scalar.
The associativity property found in the octonion-like algebra (like all Clifford algebras but unlike octonions which are non-associative) can be an important criterion for the practical applicability of an algebra. A proof for the preservation of the two seminorms under multiplication of octonion-like numbers (ZXY , ZXY ) is provided in the Appendix of this paper, using elementary linear algebra.
Because of the preservation of the seminorms under multiplication, the octonion-like algebra contains no zero-divisors using the two seminorms. It will be shown in this paper that the computation of an inverse octonion-like number involves a matrix inversion, and that the case of a singular matrix only occurs when one of the two seminorms of an octonion-like number is zero, therefore the octonion-like algebra is also a division algebra as long as the two seminorms are non-zero. Additional results on normalization of octonion-like numbers and some involutions are also presented in the paper.
The explanation for the seminormed nature of the octonion-like algebra (and lack of contradiction with Hurwitz’s theorem), the additional results related to the computation of inverse numbers in the octonion-like algebra (with a confirmation of its division algebra status using the seminorms), the additional results on normalization of octonion-like numbers and some involutions, and the developments and proof using simple linear algebra (as opposed to more abstract geometric algebra) are the contributions of this paper. The linear algebra descriptions also allow straightforward software implementations of the octonion-like algebra.
2 II. OCTONION-LIKE ALGEBRA AND MULTIPLICATION RULES
Table I presents the multiplication rules for the octonion-like algebra. From the diagonal of the table, it can be seen that in an octonion-like number:
X X(0) XeXeXeXeXeXeXe (1)1234567 (2) (3) (4) (5) (6) (7) (1)
the imaginary dimensions correspond to coefficients X(1) to X(6) , while coefficients X(0) and X(7) correspond to “scalar” or non-imaginary dimensions. We define X* as the conjugate (or reverse) of X , changing the sign of the coefficients for the imaginary dimensions in X , i.e.,X(1) to X(6) :
* X XXeXeXeXeXeXeXe(0) (1)1234567 (2) (3) (4) (5) (6) (7) (2).
Table I Table of multiplication rules in the octonion-like algebra
1st factor (row) and 2nd factor (column) of octonion-like e0 1 e1 e2 e3 e4 e5 e6 e7 number multiplication
e0 1 1 e1 e2 e3 e4 e5 e6 e7
e1 e1 -1 e3 -e2 -e5 e4 e7 -e6
e2 e2 -e3 -1 e1 e6 e7 -e4 -e5
e3 e3 e2 -e1 -1 e7 -e6 e5 -e4
e4 e4 e5 -e6 e7 -1 -e1 e2 -e3
e5 e5 -e4 e7 e6 e1 -1 -e3 -e2
e6 e6 e7 e4 -e5 -e2 e3 -1 -e1
e7 e7 -e6 -e5 -e4 -e3 -e2 -e1 1
Following these multiplication rules, the octonion-like multiplication can be detailed as:
3 ZXYXXeXeXeXeXeXeXe(0) (1)1234567 (2) (3) (4) (5) (6) (7)
Y(0) YeYeYeYeYeYeYe (1)1234567 (2) (3) (4) (5) (6) (7) XY(0)(0) XY (1)(1) XY (2)(2) XY (3)(3) XY (4)(4) XY (5)(5) XY (6)(6) XY (7)(7)
XY(1)(0) XY (0)(1)(3)(2)(2)(3)(5)(4)(4)(5)(7)(6)(6)(7)XY XY XY XY XY XY e1
XY(2)(0) XY (3)(1) XY (0)(2) XY (1)(3) XY (6)(4) XY (7)(5) XY (4)(6) XY (5)(7) e2
XY(3)(0) XY (2)(1) XY (1)(2) XY (0)(3) XY (7)(4) XY (6)(5) XY (5)(6) XY (4)(7)e3 (3).
XY(4)(0) XY (5)(1) XY (6)(2) XY (7)(3) XY (0)(4) XY (1)(5) XY (2)(6) XY (3)(7) e4
XY(5)(0) XY (4)(1) XY (7)(2) XY (6)(3) XY (1)(4) XY (0)(5) XY (3)(6) XY (2)(7) e5
XY(6)(0) XY (7)(1) XY (4)(2) XY (5)(3) XY (2)(4) X(3)(5)YXYXYe (0)(6) (1)(7) 6
XY(7)(0) XY (6)(1) XY (5)(2) XY (4)(3) XY (3)(4) XY (2)(5) XY (1)(6) XY (0)(7) e7
ZZeZeZeZeZeZeZe(0) (1)1234567 (2) (3) (4) (5) (6) (7)
From (3), we see that for dimensions eeeeee123456,,,,, substituting Xi() by Yi() and Yi() by Xi() 07i does not lead to the same value, therefore in general the octonion-like number product ZXYYX is non-commutative. Moreover, we can verify that:
(XY)*** Y X (4).
Also from (3), we can build a matrix product involving only real-valued numbers, where the matrix MX includes coefficients from the left-side octonion-like number X in the product ZXY :
ZMYrr X (5), with
T Z ZZZZZZZZ(0)(1)(2)(3)(4)(5)(6)(7) (6) r
XXXXXXXX(0) (1) (2) (3) (4) (5) (6) (7) XXXXXXXX(1)(0)(3)(2)(5)(4)(7)(6) XXXXXXXX(2)(3)(0)(1)(6)(7)(4)(5) XXXXXXXX(3) (2) (1) (0) (7) (6) (5) (4) M (7) X XXXXXXXX(4) (5) (6) (7) (0) (1) (2) (3) XXXXXXXX(5) (4) (7) (6) (1) (0) (3) (2) XXXXXXXX(6) (7) (4) (5) (2) (3) (0) (1) XXXXXXXX(7)(6)(5)(4)(3)(2)(1)(0)
4 T Y YYYYYYYY(0)(1)(2)(3)(4)(5)(6)(7) (8). r
The subscript r in ZYrr, stands for “real-valued elements only”. Alternatively, re- organizing the elements of the octonion-like product ZXY as: ZXY YX(0)(0) YX (1)(1) YX (2)(2) YX (3)(3) YX (4)(4) YX (5)(5) YX (6)(6) YX (7)(7)
YX(1)(0) YX (0)(1) YX (3)(2) YX (2)(3) YX (5)(4) YX (4)(5) YX (7)(6) YX (6)(7) e1
YX(2)(0) YX (3)(1) YX (0)(2) YX (1)(3) YX (6)(4) Y (7)XYXYXe(5) (4) (6) (5) (7) 2 (9) YX(3)(0) YX (2)(1) YX (1)(2) YX (0)(3) YX (7)(4) YX (6)(5) YX (5)(6) YX (4)(7) e3
YX(4)(0) YX (5)(1) YX (6)(2) YX (7)(3) YX (0)(4) YX (1)(5) YX (2)(6) YX (3)(7) e4
YX(5)(0) YX (4)(1) YX (7)(2)YX(6)(3) YX (1)(4) YX (0)(5) YX (3)(6) YX (2)(7) e5
YX(6)(0) YX (7)(1) YX (4)(2) YX (5)(3) YX (2)(4) YX (3)(5) YX (0)(6) YX (1)(7) e6
YX(7)(0) YX (6)(1) YX (5)(2) YX (4)(3) YX (3)(4) YX (2)(5) YX (1)(6) YX (0)(7) e7 we can build a 2nd matrix form involving only real-valued numbers, where this time the
matrix PY includes coefficients from the right-side octonion-like number Y in the product ZXY :
ZPXrr Y (10), with:
YYYYYYYY(0) (1) (2) (3) (4) (5) (6) (7) YYYYYYYY(1)(0)(3)(2)(5)(4)(7)(6) YYYYYYYY(2)(3)(0)(1)(6)(7)(4)(5) YYYYYYYY(3)(2)(1)(0)(7)(6)(5)(4) P (11) Y YYYYYYYY(4)(5)(6)(7)(0)(1)(2)(3) YYYYYYYY(5) (4) (7) (6) (1) (0) (3) (2) YYYYYYYY(6) (7) (4) (5) (2) (3) (0) (1) YYYYYYYY(7)(6)(5)(4)(3)(2)(1)(0)
T X XXXXXXXX(0)(1)(2)(3)(4)(5)(6)(7) (12). r
As a special case of the previous detailed formulation for ZXY , ZXX * with the conjugate X* as in (2) can be detailed as:
5 * XX X(0) XeXeXeXeXeXeXe (1)1234567 (2) (3) (4) (5) (6) (7)
XXeXeXeXeXeXeXe(0) (1)1234567 (2) (3) (4) (5) (6) (7) XX(0)(0) XX (1)(1) XX (2)(2) XX (3)(3) XX (4)(4) XX (5)(5) XX (6)(6) XX (7)(7)
XX(1)(0) XX (0)(1)XX(3)(2) XX (2)(3) XX (5)(4) XX (4)(5) XX (7)(6) XX (6)(7) e1
XX(2)(0) XX (3)(1) XX (0)(2) XX (1)(3) XX (6)(4) XX (7)(5) XX (4)(6) XX (5)(7) e2 (13). XX(3)(0) XX (2)(1) XX (1)(2) XX (0)(3) XX (7)(4) XX (6)(5) XX (5)(6) XX (4)(7)e3
XX(4)(0) XX (5)(1) XX (6)(2) XX (7)(3) XX (0)(4) XX (1)(5) XX (2)(6) XX (3)(7) e4
XX(5)(0) XX (4)(1) XX (7)(2) XX (6)(3) XX (1)(4) XX (0)(5) XX (3)(6) XX (2)(7) e5
XX(6)(0) XX (7)(1) XX (4)(2) XX (5)(3) XX (2)(4) X (3)XXXXXe (5) (0) (6) (1) (7) 6
XX(7)(0) XX (6)(1) XX (5)(2) XX (4)(3) XX (3)(4) XX (2)(5) XX (1)(6) XX (0)(7) e7
It can be noted that all the imaginary dimensions in the octonion-like number XX* have zero coefficients, so (13) reduces to:
XX* XX(0)(0) XX (1)(1) XX (2)(2) XX (3)(3) XX (4)(4) XX (5)(5) XX (6)(6) XX (7)(7)
XX(7)(0) XX (6)(1) XX (5)(2) XX (4)(3) XX (3)(4) XX (2)(5) XX (1)(6) XX (0)(7) e7 (14). 73 Xi2() 2 Xi ()X(7 i ) 2 X (0) X (7) e 7 ii01
Unlike the general octonion-like product XY , the product XX* is commutative ( XX** X X and ()XX** Y Y () XX ), as can be observed from (9), (13) and (14).
The fact that the octonion-like product XX* has non-zero coefficients only for dimensions T T e0 1 and e7 has a corresponding result for the matrix products MMXX and PPXX: they are the sum of a diagonal matrix and an anti-diagonal matrix, whose non-zero elements are
located at the positions of results with dimensions e0 1 and e7 in Table 1, and whose
values are obtained from the e0 1 and e7 components in (14):
6 ab000000 0ab 0000 0 00ab 00 00 000ab 000 MMTT PP (15) XX XX 000 ba 000 00 ba 00 00 0 ba 0000 0 ba000000 with
7 aXi2() XXT rr i 0 (16) 63 b XXi()iXXXiXX(7 ) 2(0)(7) 2 () Xi (7) 2(0 ) (7)(XJXT ) rr ii11 and
00000001 00000010 00000100 00001000 J (17). 00010000 00100000 01000000 10000000
* It can be noted that JXr is a flipped version of the conjugate octonion-like number X coefficients:
T JX XXXXXXXX(7)(6)(5)(4)(3)(2)(1)(0) (18). r
T We can then write MMXX compactly:
TT T MMXX ()(()) XXrr I+X r JXJ r (19).
These results will be useful for the proof to be presented in the Appendix.
7 To complete this section, we also present a result using the matrix MX to compute the integer power of an octonion-like number:
n XXX (20) n
XMnnn MXMXSSX111 (21) rrrrXX X n1
1 for n 1 integer, and where MSSX assumes that the diagonalization of the matrix
MX is possible, with eigenvalues in the diagonal matrix and corresponding eigenvectors in the columns of S.
III. SEMINORMS FOR THE OCTONION-LIKE ALGEBRA
Consider the following non-negative scalar expression resembling the previous expression for the product XX* in (14). This expression is the square of two seminorms X :
2 X XX(0)(0) XX (1)(1) XX (2)(2) XX (3)(3) XX (4)(4) XX (5)(5) XX (6)(6) XX (7)(7) XX(7) (0) XX (6) (1) XX (5)(2)(4)(3)(3)(4)(2)(5)(1)(6)(0)(7)XX XX XX XX XX (22). 76 73 Xi22()X () iX(7 i ) 2 X (0) X (7) Xi () 2()2(0)(7XiXi(7 )) X X ii01 ii01
There is a seminorm with 1 and a seminorm with 1. Note that with this definition we have XX* , with the conjugate X* as in (2). Eq. (22) corresponds to seminorms
and not norms because X 0 does not imply that X 0 . These seminorms can also be expressed in compact vector form:
2 TT XXXXJXrr r() r (23).
It is proven in the Appendix that for these two seminorms the octonion-like algebra is a seminormed algebra, i.e., ZXY , ZXY .
8 7 In the seminorms of (22) the term Xi2() is always larger or equal to the amplitude of i0 3 the other terms 2()XiXi(7 ) 2(0) X X (7) . If Xi2() 0 i 7 has either some i 1
7 dominant values or is sparsely populated, then the term Xi2() becomes dominant and i0 the seminorms become similar to the Euclidian norm. The seminorms in (22) have a value of zero only when the following conditions are met:
all the terms Xi() which are non-zero have the same magnitude Xi(); for each non-zero term Xi() there is a corresponding non-zero term Xi(7 ) 0 i 7
3 all the non-zero products in 2()XiX(7 i ) 20 X ( ) X (7) have the same polarity. i 1
Since the octonion-like algebra is a seminormed algebra, the algebra has no zero-divisors using the two seminorms in (22). For example, in the simple octonion-like product
(1ee77 )(1 ) 1 ee 77 1 1 0 with (1ee77 ) (1 ) 0 0 , either (1 e7 ) has a seminorm of zero (if seminorm with 1 is used) or (1 e7 ) has a seminorm of zero (if seminorm with 1 is used), so neither (1 e7 ) or (1 e7 ) is a zero divisor using any of the two seminorms.
IV. INVERSE NUMBERS IN THE OCTONION-LIKE ALGEBRA
The octonion-like algebra allows computation of inverses (and thus to perform divisions). However, computing the inverse of an octonion-like number (or performing divisions in general) requires a matrix inversion, as shown below. Consider an inverse Xi with the same form as the inverse used for complex number, quaternion or octonion algebras:
X* XX1 (24). i 2 X
Using the multiplication rules for octonion-like numbers and the previous result for XX* in (14), for a right-side inverse we obtain:
9 73 Xi2() 2 XiX () (7 i ) 2 X (0) X (7) e * 7 XX XX ii01 (25), i 22 XX and with the definition of the seminorms for octonion-like numbers in (22), this becomes:
73 Xi2() 2 XiX () (7 i ) 2 X (0) X (7) e * 7 XX XX ii01 1 (26). i 27 3 X Xi2()2 XiX ()(7)2(0)(7) i X X ii01
We see that this is not the right-side inverse of an octonion-like number such that XXi 1 X* X , although XX X X 1, i.e., it is a correct right-side inverse if only a i 22 XX seminorm is considered.
A right-side inverse can be obtained by writing the octonion-like product XXi 1 in the matrix form introduced earlier in (5):
T MX 10000000 (27) X ir, with
T X XXXXXXXX(0)(1)(2)(3)(4)(5)(6)(7) (28). iriiiiiiii,
Therefore, we have an 8 by 8 linear set of equations to solve (which can possibly be implemented more efficiently making use of the structure in MX , but this is not 1 investigated here). Or, if we explicitly use the matrix inverse MX :
T XM 1 10000000 (29). ir, X
Similarly, the solution for a left-side inverse XXi 1 can be directly obtained by writing the octonion-like product XXi 1 with the matrix form introduced earlier in (10):
T PX 10000000 (30) X ir,
T XP 1 10000000 (31). ir, X
10 However, the solution Xi for the right-side inverse XXi 1 is the same as for the left-side inverse XXi 1, as any product XY YX is commutative when the result only has e0 and e7 components in (9).
To evaluate when MX or PX can be singular, the eigenvalues of MX or PX are:
01,(0)(7)XX iX(1)222 2 XX (1) (6) X (2) 2 XX (2) (5) X (3) 2 XX (3) (4) X (4) 222 X (5) X (6)
23,(0)(7)XX iX(1)222 2 XX (1) (6) X (2) 2 XX (2) (5) X (3) 2 XX (3) (4) X (4) 222 X (5) X (6) (32) 45,(0)(7)XX iX(1)2 2 XX (1) (6) XXXXXXXXX(2)22222 2 (2) (5) (3) 2 (3) (4) (4) (5) (6)
67,(0)(7)XX iX(1)2(1)(6)222 XX X (2)2(2)(5) XX X (3)2(3)(4) XX X (4) 222 X (5) X (6) and their magnitude is:
76 ,,,XXTT XJX Xi2 () Xi ()Xi(7 ) 2(0)(7 X X ) a b 0123rr r r ii01 (33). 76 , , ,XXTT - XJX Xi2 () Xi ()Xi(7 ) 2 X (0) X (7) a b 4567rr r r ii01
The magnitude of the eigenvalues in (33) is therefore the same as the two seminorms defined 76 in (22) and it is non-negative with Xi2() Xi ()Xi(7 )7) 2 X (0) X ( . The cases with ii01 zero eigenvalues (and singular MX or PX ) only occur if one of the two seminorms in (22) is zero, and the conditions required for this to happen have been described in the previous section. Therefore, since all octonion-like numbers with non-zero values for the two seminorms in (22) have an inverse, the octonion-like algebra is a division algebra as long as the two seminorms are non-zero.
The following property also applies for the inverse of octonion-like numbers:
(XY)111 Y X (34), which is easily verified from:
11 XY() Y11 X XYY 11 X 1 (35) and therefore (XY)111 Y X .
V. NORMALIZATION OF OCTONION-LIKE NUMBERS AND INVOLUTIONS
We are interested to find an octonion-like number Xd such that its inverse can be used for a commutative multiplicative normalization:
11*11** (XXdd)( XX ) ( X dd X )( X X ) X nn X 1 (36).
With the normalised number Xn , the inverse number becomes the conjugate:
1* XX=Xni, n n (37).
* and XXnn1 for the seminorms in (22). Using the previously introduced inverse
T T number, i.e., XXM11 110000000 or XXP11110000000 , dr,, ir X dr,, ir X would be an obvious solution for this problem, but this would lead to the same normalized value Xn 11e0 for any number X and would be of limited use. Another solution for 1 Xd is described next, which will lead to more interesting involution results.
First we note from the definition of ZXY in (9) that if either X or Y has non-zero components only for the real (scalar, non-imaginary) dimensions ee07, , then ZXY becomes commutative. We also note that for an octonion-like number X with non-zero components only for dimensions ee07, , MX in (7) becomes the sum of a diagonal and an 1 anti-diagonal matrix, therefore MX also becomes the sum of a diagonal and an anti- diagonal matrix, and consequently the inverse number computed with T XM11 10000000 is non-zero only for components ee, . Therefore, a simple way ir, X 07 1 to ensure that a product with Xd is commutative is to constrain Xd to be non-zero only for the ee07, components, i.e., Xd ()Ae07 Be . This leads to:
11*11**11**11 (XXd)( XX d ) XXX d ( d ) X XXXX dd XXXX dd 1 (38)
12 73 XX XX*2Xie() 2 Xi ()X(7iXe ) 2 X (0) (7) (39) dd07 ii01
73 ()()()2()2(0Ae Be Ae Be X2 i eXe iX(7i))7) X X ( (40) 0707 0 7 ii01
7 AB22 Xi 2() (41) i 0
3 22()2(0)(7)AB X iXi(7 ) X X (42). i 1
Isolating A from (42) and inserting in (41):
2 73 422 44BBXi ()2() XiX(7 iX ) 2(0)( X 7) 0 (43) ii01
2 73 22 44BBXi ()2()2(0)() XiXi(7 )70 XX (44) ii01
with BB (both roots are possible and lead to a correct Xd solution, i.e., Xd is not unique). Solving for B :
22 77 3 22 4()16()16Xi Xi2() XiXX (7 i)(7) 2(0) X ii00 i 1 B (45) 8 where we use the positive root to guarantee that B is always positive (the negative root only guarantees non-negativity). Solving for BBA,, , we find a commutative number
*11*11* Xd Ae07 Be for which Xnn X()()()()1 XX d XX d X d X X d X .
For octonion-like numbers Xˆ whose non-zero components are only imaginary components
eeeeee123456,,,,, (“pure” numbers, i.e., real components are zero: XX(0) 0, (7) 0 ), since XXˆˆ* the normalization becomes:
* XXˆˆ11 XX ˆˆ XX ˆˆ *1*111 XX ˆˆ XX ˆˆ XX ˆˆ XX ˆˆ XX ˆˆ 11e (46) dd nnd d dd nn0
ˆˆ XXnn11e0 (47).
13 ˆ ˆ This result may be useful for the development of polar-like representations. Like X , Xn also only has imaginary components eeeeee123456,,,,, with non-zero values, as the product ˆ 1 of a commutative octonion-like number Xd (having non-zero coefficients only in dimensions ˆ ee07, ) with a number X (having non-zero coefficients only in dimensions eeeeee123456,,,,, ˆ ) also leads to a number Xn with non-zero coefficients only in dimensions eeeeee123456,,,,, . This can be verified from the definition of ZXY in (9).
We define next the following operation with the same form as rotations in quaternion and octonion algebras:
XUXUU 1 (48). ˆ If Un is an arbitrary imaginary and “unit” octonion-like number, i.e., has been normalized ˆˆ such that UUnn1 , (48) becomes:
Uˆ n ˆˆ1* ˆˆ ˆˆ X Unn XU U nn XU U nn XU (49), which is an involution: ˆ ˆ ˆ ˆ ˆˆ ˆˆ U(n UXU)U n n n UUXUU nn nn 11 X X (50). ˆ The unit eeeeee123456,,,,, are specific cases of such imaginary and “unit” numbers Un .
Although not imaginary, the unit ee07, numbers also lead to involutions for
Un Un 1* XUXU nnor XUXUUXUnn nn. But in general normalized non-imaginary
Un 1* numbers Un aren’t involutions under XUXUUXUnn nn.
For the definition in (48) we can also derive the following results and properties:
XXUU/ a a real (51)
UU11 U U U XXU dd X X n (52)
(XY)UUU X Y (53)
* XY YX X YXY (54)
XXYZ () Z Y (55)
* XXUUU()** ( X ) (56).
14 For the specific case of the unit octonion-like numbers eeeeeeee01234567(1),,,,,,, , we also have:
ee XXii7 03i (57)
eeeeeee e X+X0356712 X X X 4 X X X (58) ee03ee12 2X+2X 2 X 2 X 8Xe (0)07 8 Xe (7)
eee e e eee e e e e ()(2222)4XXXXXXXX035670312 4 ** X X12 X X X (59).
Finally, using (48) with non-zero U generates an alternative basis:
U 1 1 XUXUU(xxexexexexexexe011223344556677) U 1111111 xxexexexexexexe011223344556677UU UU UU UU UU UU UU (60) UUUUUUU xxexexexexexexe011223344556677
U where the resulting ei terms still obey the rules of the octonion-like algebra:
U U UU UUUUUU U ee001 ee77 eeii11 i 6 eeeeee123456 1 eii 10 7 (61).
VI. CONCLUSION
In conclusion, the octonion-like algebra (even subalgebra of Clifford algebra Cl40, () ) is an associative seminormed division algebra over , which does not contradict Hurwitz’s theorem. It is in general a non-commutative algebra (just like quaternions and octonions), although some operations such as number inverse, normalization or multiplication with numbers having no imaginary components are commutative. In addition to proving the seminormed nature of the algebra, this paper also provided results for the computation and existence of inverse numbers, as well as for number normalization and for some involutions. The results were developed using elementary linear algebra, which makes them easily accessible to many readers and can facilitate software implementations. Considering the isomorphism with lower dimensional normed spaces, it is unclear at this stage for which applications the use of the octonion-like associative seminormed division algebra over (and its seminorms) can provide benefits. But some applications have already been suggested in [6].
15 VII. ACKNOWLEDGEMENTS
The authors would like to thank Dr. Joy Christian and Dr. Richard D. Gill for helpful comments and clarifications on different versions of this paper.
VIII. REFERENCES
[1] J. Christian, "Eight-dimensional Octonion-like but Associative Normed Division Algebra," arXiv:1908.06172v8 [math.GM] , November 2020.
[2] J. Christian, "Quantum Correlations Are Weaved by the Spinors of the Euclidean Primitives," R. Soc. Open Sci., , Vols. 5, 180526 , 2018.
[3] "Hurwitz's theorem (composition algebras) (Wikipedia)," [Online]. Available: https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras).
[4] R. D. Gill, "Does Geometric Algebra Provide a Loophole to Bell’s Theorem?," Entropy, Vols. 22, 61; doi:10.3390/e22010061, 2020.
[5] "Split-biquaternion (Wikipedia)," [Online]. Available: https://en.wikipedia.org/wiki/Split-biquaternion.
[6] A. N. Lasenby, "A 1d Up Approach to Conformal Geometric Algebra: Applications in Line Fitting and Quantum Mechanics," Adv. Appl. Clifford Algebras , vol. 30:22, 2020.
IX. APPENDIX: PROOF FOR PRESERVATION OF SEMINORMS PRODUCT UNDER OCTONION-LIKE NUMBER MULTIPLICATION
We need to show that:
2222 ZXYXY (62),
16 where XYZ,, are octonion-like numbers as before. With the two seminorms for octonion- like numbers in (22), this is equivalent to:
73 Zi2() 2 Zi ()Zi(7 ) 2 Z (0)(7) Z ii01 (63), 73 73 2 2 Xi( ) 2 Xi ( )Xi(7 )(7)(7 2 X (0) X (7) Yi ( ) 2 Yi ( ) Yi 2YY (0 ) ) ii01 ii 01
2 TT with 1. In matrix form, since XXXXJXrr r() r we have:
ZZTT Z() JZ XX T X T ( JX ) YY T Y T ( JY ) rr r r r r r r r r r r (64). YYTT Y() JY XX TT X () JX rr r r rr r r
Since ZMYrr X , we can write:
YMMYTT YM TT() JMY rrrrXX X X (65) TT TT T T T T YYXXrrrr YYX rrr() JX r Y r () JYXX r rr Y r ()() JYX r r JX r and
MMYTT MJMY XXrr X X (66). TT TT YXXrrr() YXJX rr ( r ) JYXX rrr () JYXJX rr ( r )
This condition holds true if the two following conditions are true:
TTT MMYXXrrrrrrr Y() XX JYXJX ( ) (67)
TTT MJMYXXrrrrrrr Y() XJX JYXX () (68).
The 1st condition is easily shown using the previous result MMTT ()(()) XX I+X T JXJ XX rr r r :
TTT MMYXXrrrrrr (( XX ) I+X ( ( JXJY )) ) TT ()(XXrr Y+XJXJY r r r ) r (69). TT YXXrrr() +JYXJX rr ( r )
nd TT For the 2 condition, first we show that MJMXX JMM XX, making use of JJ I :
TTT MJMYXXrrrr Y()() XJX JYXX rrr (70)
17 TTT JMXX JM Yrrrrrrr JY() X JX JJY () X X TT JYXJXrr()() r Y rrr XX (71) T MMYXXr
TT JMXX JM M XX M (72)
TT JJMXX JM JM XX M (73)
TT MJMXX JMM XX (74).
nd TT T Then the 2 condition is easily verified, again using MMXX ()(()) XXrr I+X r JXJ r :
MJMYTTTT JMMY J((()) XX)I+X JXJY XXrrrrrrr XX (75). TT JYrrr()( X X + Y rr X JX r )
18