LOOPS EMBEDDED IN GENERALIZED CAYLEY ALGEBRAS OF DIMENSION 2 r , r ≥ 2

Every Cayley algebra of dimension 2r , r ≥ 2, contains an embedded invertible loop of order 2r+1 generated by its basis. Such a loop belongs to a class of non-abelian invertible loops that are flexible and power-associative. 2000 Mathematics Subject Classification. 20N05.


Introduction. Every finite-dimensional algebra
A over a field F can be defined by a multiplication table of its basis E n = {e 1 ,...,e n }.Such a table can be expressed by a matrix M r (E n ) = (m ij ), i, j = 1,...,n, called a multiplication matrix or ⊗-matrix, where m ij = e i ⊗e j = n k=1 γ k ij e k , γ k ij ∈ F are its structure constants, and e i , e j , e k ∈ E n .By a suitable choice of structure constants, it is possible to construct algebras with desired properties.
There is a class of real algebras called Cayley algebras of dimension n = 2 r , where r ≥ 2 [2].This class includes the classical Cayley-Dickson algebras H (quaternions) and O (octonions) [3] as well as the sedenions S. In this note, we show that the basis of such an algebra A forms a non-abelian invertible loop of order 2 r +1 , called a Cayley loop, that is flexible and power-associative.Moreover, we also indicate how the idea of the ⊗-matrix can be used in the construction of special algebraic structures (like the group of Dirac operators in quantum electrodynamics).
2. The ⊗-matrix of a Cayley algebra.Consider the ⊗-matrix M 3 (E 8 ) = (m ij ) shown in Figure 2.1 which defines the algebra of Cayley numbers (or octonions) O [2].If we separate the sign coefficients (or structure constants) z ij of the entries of M 3 into another matrix Z 3 (E 8 ) (Figure 2.2(a)), then the resulting matrix S 3 (E 8 ) can be seen to be the Cayley table of the Klein group (E 8 , •) of order n = 8 shown in Figure 2.2(b).This group is isomorphic to the group where C 2 is the cyclic group of order 2.
The decomposition of the ⊗-matrix M 3 (E 8 ) into two other matrices Z 3 (E 8 ) and S 3 (E 8 ), therefore, shows that the algebras O, H, and C are defined by ⊗-matrices of the form ᏹ r (E s ) = ᐆ r (E s ) r (E s ), where is Hadamard multiplication, that is, if ᐆ r = (z ij ) and r = (e ij ) are matrices of the same dimension, then their Hadamard product is ᏹ r = (m ij ), where m ij = z ij •e ij , and • is some binary operation.
, where e ij = e i •e j = e v , is the structure matrix of the Klein group (E 8 ; •) C 3 2 of order 8.For notational simplicity, we have used ± to represent ±1 and the subscript v to represent the element e v ∈ E 8 .
We now formally define the matrices ᐆ, , and ᏹ as follows.
Thus, an m ×n sign matrix [1] is one whose entries are elements of the number set F = {+1, −1} ∈ R. Therefore, they satisfy the following composition rule: This rule shows that the number set F = {+1, −1} is closed under the operation • of multiplication; they form a group (F , •) isomorphic to the cyclic group C 2 of order 2.
Definition 2.2.Let (E s , •) be a finite binary system (like a quasigroup, group or a loop) of order s, where E s = {e 1 ,...,e s }.The matrix (E s ) = (e ij ), where e ij = e i •e j , is called the structure matrix (or Cayley table) of (E s , •).Every finite binary system of order s is completely defined by its structure matrix (or Cayley table) which is a listing of all the s 2 possible binary products of its s elements.In the case of finite quasigroups, loops, and groups, their Cayley tables form Latin squares.
Definition 2.3.Let (E s , •) be a binary system of order s, where E s = {e i | i ∈ I} and I = {1,...,s}, and let r (E s ) = (e ij ) be its structure matrix, where e ij = e i • e j for all i, j ∈ I.
is called the multiplication matrix or ⊗-matrix of E s , where is Hadamard multiplication such that for all i, j ∈ I, and the operation • is called sign multiplication.
In Definition 2.3 of the ⊗-matrix, we introduced the operation ⊗ in terms of the operation • of sign multiplication in the expression e i ⊗ e j = z ij • e ij .This operation • simply attaches a sign z ij (+ or −) to the left of the symbol e ij .If we take ᐆ r to be a sign matrix whose entries z ij are the numbers +1 and −1; and r to be a structure matrix whose entries e ij are elements of a set E s of vectors, then we can take the operation • to be ordinary scalar multiplication so that the product e i ⊗e j = z ij •e ij will be a vector.This would be the case if ᏹ r (E s ) is the ⊗-matrix of a finite-dimensional real algebra whose basis is E s .This is exemplified by the octonions which we discussed above.

The Cayley loops.
It follows from Definition 2.3 that if ᏹ r (E s ) is the ⊗-matrix of a real algebra A, then the operation ⊗ is closed over A but not over E s because of the sign coefficient z ij in its defining equation e i ⊗ e j = z ij • (e i • e j ).Thus, if z ij = −1, then −(e i •e j ) ∉ E s .However, if we take the larger set Ᏹ = {±e i | i ∈ I} of order σ = 2s, where +e i ≡ (+1)e i = e i and −e i ≡ (−1)e i , then the operation ⊗ will be closed over Ᏹ.This means that the system (Ᏹ, ⊗) is a groupoid embedded in the algebra A. Such a groupoid will be called a ⊗-system.
Consider once more the octonion algebra O.This is defined by the ⊗-matrix 2.1.For this case, (E 8 , •) is the Klein group of order s = 8, and the operations ⊗, •, and the matrix Z 3 satisfy the following basic relations: Equations (3.2) define the basic properties of the entries of the sign matrix Z 3 while (3.3), on the other hand, define the basic properties of the products elements of E 8 under the operation ⊗ Any real algebra (like the octonions O and sedenions S) defined by a ⊗-matrix of the form ᏹ r (E s ) = ᐆ r (E s ) r (E s ), satisfying (3.1), (3.2), and (3.3),where (E s , •) C r 2 will be called a Cayley algebra of dimension s = 2 r , r ≥ 2. In such an algebra, the set Ᏹ = {±e i | i = 1,...,s} and the operation ⊗ form an embedded non-abelian ⊗system (Ᏹ, ⊗) that is an invertible loop (a loop in which every element has a unique two-sided inverse), where δ i e i ⊗δ j e j = (δ i δ j )[z ij •(e i •e j )] and δ i ,δ j ∈ F .This form of the composition rule is implied by (3.1).In the case of the octonions O, the ⊗-system (Ᏹ, ⊗), where Ᏹ = {±e i | i = 1,...,8}, forms a non-abelian invertible loop of order 16 called the octonion loop.In general, we have the following theorem.3), it follows that (Ᏹ, ⊗) is a nonabelian groupoid of order 2 r +1 with an identity e 1 .Moreover, (Ᏹ, ⊗) is invertible, that is, every element e x ∈ Ᏹ has a unique inverse e −1  x ∈ Ᏹ.Thus e −1 1 = e 1 and e −1 x = −e x since e x ⊗ (−e x ) = −e x ⊗ e x = e 1 for all x ≥ 2. Similarly, every element −e x ∈ Ᏹ has a unique inverse (−e x ) −1 = e x ∈ Ᏹ.To prove that (Ᏹ, ⊗) is an invertible loop, it is therefore sufficient to show that every linear equation has a unique solution.By (3.1), the product of any two elements in (Ᏹ, ⊗) is determined primarily by the product (e i • e j ) in (E s , •).Since (E s , •) is a group, then every linear equation has a unique solution.This, together with (3.1) and (3.3), imply that this is also true for (Ᏹ, ⊗).Therefore, (Ᏹ, ⊗) is an invertible loop.By definition, every Cayley algebra A of dimension 2 r is defined by a ⊗-matrix satisfying (3.1), (3.2), and (3.3).Therefore, it follows from Theorem 3.1 that its embedded ⊗-system (Ᏹ, ⊗) is a Cayley loop.Thus, the octonion loop generated by the basis of the octonion algebra is a Cayley loop.Similarly, the loop generated by the basis of the sedenion algebra is also a Cayley loop.
The Cayley loop (Ᏹ, ⊗) can be explicitly expressed in terms of the matrix ᏹ r (E s ) = ([m r ] ij ) as follows.Let (Ᏹ) be the structure matrix of (Ᏹ, ⊗).Partition (Ᏹ) into four blocks Ᏹ pq , p, q = 1, 2, and let Ᏹ 11 = Ᏹ 22 = ᏹ r (E s ) and Ᏹ 12 = Ᏹ 21 = −ᏹ r (E s ), where −ᏹ r (E s ) = (−[m r ] ij ).Then we can simply write (Ᏹ) = (Ᏹ pq ).The structure matrix (Ᏹ) of (Ᏹ, ⊗) is shown below in block form in terms of the matrix ᏹ r (E s ) Every Cayley loop or ⊗-system (Ᏹ, ⊗) can be expressed in this matrix form (Ᏹ).This matrix clearly shows that (Ᏹ, ⊗) is an invertible loop and it can be used as an alternative proof of Theorem 3.1.Many important invertible loops and groups have this structure.

Construction of Cayley loops.
The foregoing considerations show that we can construct special loops by means of ⊗-matrices.As an illustration, consider the case of the 4 It can be shown [2] that if Z r ,v is any n × n sign matrix satisfying (3.2), then there are exactly |Z r ,v | = 2 µ matrices of this form, where µ = n−1 i=2 (n − i).Since n = 4, then we find that µ = 3. Hence there are |Z 2,v | = 2 3 = 8 possible 4 × 4 Z 2,v matrices so that v = 1,...,8.These eight sign matrices are shown in Figure 3.1.It can be shown that the ⊗-matrices M 2,3 and M 2,7 generate loops both of which are isomorphic to the quaternion group.The other six ⊗-matrices, on the other hand, generate non-associative finite invertible loops (NAFILs) that are isomorphic to each other.
Although the idea of the ⊗-matrix ᏹ r (E s ) = ᐆ r (E s ) r (E s ) is based on the multiplication matrix of the Cayley algebras, Definition 2.3 is not restricted to these algebraic systems.Such a matrix can therefore be used to construct not only Cayley loops but also other structures (like the group of Dirac operators in quantum electrodynamics [1]) which we call ZSM loops.Starting with a given group (E s , •), new systems can (3.3).Note that M 2,3 and M 2,7 are transposes and that both generate Cayley loops isomorphic to the quaternion group be formed by means of ᐆ-matrices.The given group is thus the substratum of such a system, while the ᐆ-matrix determines its special properties.Proof.By Theorem 3.1, (Ᏹ, ⊗) is a non-abelian invertible loop.To prove that it is flexible, let e i ,e j ∈ Ᏹ.Then the following identity (called the flexible law) must be satisfied: e i ⊗ e j ⊗ e i = e i ⊗ e j ⊗ e i (3.5) for all e i ,e j ∈ Ᏹ.Clearly, this is trivially satisfied if i, j = 1 and also if e i and e j are inverses.By (3.3), if i ≠ j, i, j ≥ 2, the left side of this identity can be written as e i ⊗ (e j ⊗ e i ) = −(e j ⊗ e i ) ⊗ e i .But (e j ⊗ e i ) = −(e i ⊗ e j ) so that we have −(e j ⊗ e i ) = (e i ⊗ e j ).Therefore, it follows that e i ⊗ (e j ⊗ e i ) = (e i ⊗ e j ) ⊗ e i ; and hence (Ᏹ, ⊗) is flexible.To prove that (Ᏹ, ⊗) is power-associative, we must show that it satisfies the following two equations: e i ⊗ e If r = 2, then there exist two Cayley loops of order n = 8, one of which is an NAFIL while the other is a group (the quaternion group).All Cayley loops, whether associative or nonassociative, are non-abelian, flexible, and power-associative.Some Cayley loops (like the octonion loop) are Moufang, and hence also alternative [2] and IP.Others (like the sedenion loop) are alternative and IP but not Moufang.

Properties of
Although all basic properties of a generalized Cayley algebra are determined by the embedded Cayley loop generated by its basis, not all properties of the loop are satisfied by the algebra.For instance, the sedenion loop that defines the sedenion algebra is alternative but the sedenion algebra is not.
It is easy to show that the elements e 1 , −e 1 commute and associate with the elements e i ∈ Ᏹ.This implies that the set {e 1 , −e 1 } is the center of the Cayley loop (Ᏹ, ⊗).
It would be interesting to find out if the inner mappings of Cayley loops are automorphisms.This, and other interesting questions, are the subject of our present studies.

Summary.
The class of Cayley algebras of dimension 2 r , where r ≥ 2, is a generalization of the classical Cayley-Dickson algebras.Such an algebra is defined in terms of its basis E s = {e 1 ,...,e s } by a ⊗-matrix of the form ᏹ r (E s ) = ᐆ r (E s ) r (E s ) = ([m r ] ij ), where [m r ] ij = e i ⊗ e j = z ij • e ij = z ij • (e i • e j ), in which ⊗ satisfies (3.1), (3.2), and (3.3).By forming the set Ᏹ = {±e i | i ∈ I} of order σ = 2 r +1 , we showed that the system (Ᏹ, ⊗) is a non-abelian invertible loop, called a Cayley loop, that is flexible and power-associative.
Although all properties of a generalized Cayley algebra are determined by its Cayley loop, not all properties of the loop are satisfied by the algebra.

Figure 2 . 1 .
Figure 2.1.The ⊗-matrix M 3 (E 8 ) = (m ij ), where m ij = e i ⊗ e j = z ij e k which defines the real division algebra A 3 of dimension n = 8 isomorphic to the Cayley numbers (or octonions) O.To simplify the notation for the entries m ij , we have set z ij k ≡ z ij e k , where z ij = ±1 and k = 1,...,8.

Figure 3 . 1 .
Figure 3.1.Eight possible Z-matrices Z 2,v that can be used to form eight matrices M 2,v [as shown in Figure 3.2] satisfying (3.2).Note that the matrices in the top row are the transposes of those in the bottom row.Thus, Z 2,3 and Z 2,7 are transposes, etc.

Figure 3 .
Figure 3.1 shows the eight matrices Z 2,v which, together with the submatrix S 2 shown in Figure 2.2(b), are used to form the eight ⊗-matrices M 2,v shown in Figure 3.2.These matrices, in turn, can be used to construct eight Cayley loops of order σ = 8 whose structure matrices have the form given by (3.4).It can be shown that the ⊗-matrices M 2,3 and M 2,7 generate loops both of which are isomorphic to the quaternion group.The other six ⊗-matrices, on the other hand, generate non-associative finite invertible loops (NAFILs) that are isomorphic to each other.Although the idea of the ⊗-matrix ᏹ r (E s ) = ᐆ r (E s ) r (E s ) is based on the multiplication matrix of the Cayley algebras, Definition 2.3 is not restricted to these algebraic systems.Such a matrix can therefore be used to construct not only Cayley loops but also other structures (like the group of Dirac operators in quantum electrodynamics[1]) which we call ZSM loops.Starting with a given group (E s , •), new systems can