Central Products of Cayley-Dickson Loops

Technical Summary: Central Products of Cayley-Dickson Loops

Problem Statement
This paper investigates the structural properties of central products of Cayley-Dickson loops, specifically focusing on the behavior of commutators and the conditions under which such products are isomorphic. The authors aim to distinguish the behavior of these non-associative structures from that of groups. While groups exhibit rigid constraints on commutativity (e.g., the commutativity degree of a non-abelian group cannot exceed $5/8$), the paper explores whether Cayley-Dickson loops, which are di-associative (every two elements generate an associative subgroup), can exhibit significantly different probabilistic behaviors regarding commutativity and associativity. Additionally, the paper addresses the uniqueness of decomposition in central products, asking whether an isomorphism between two central products of $n$-fold Cayley-Dickson loops implies a term-wise isomorphism of the constituent loops.

Methodology
The authors utilize the recursive construction of Cayley-Dickson algebras $(\gamma_1, \dots, \gamma_n)_F$ over a field $F$ (where $\text{char}(F) \neq 2$) to define $n$-fold Cayley-Dickson loops $(\gamma_1, \dots, \gamma_n)_Z$ over a subgroup $Z \subseteq F^\times$. These loops are defined as multiplicative subloops generated by $Z$ and the algebra's generators.

The core methodology involves:

1. Algebraic Quotients: Utilizing the fact that for any $n$-fold Cayley-Dickson loop $D$, the quotient $D/Z$ is isomorphic to the additive group $(\mathbb{Z}/2\mathbb{Z})^n$. This allows the authors to treat the loops as vector spaces over $\mathbb{F}_2$ for the purpose of counting arguments.
2. Rank Definition: Defining the "rank" of an element in a central product $A = D_1 * \dots * D_m$ based on the number of constituent loops from which its generators are drawn.
3. Probabilistic Counting: Calculating the size of centralizers $C_A(x)$ and the probability that random commutators are trivial by analyzing the distribution of ranks and the properties of the central product modulo the center $Z$.
4. Inductive Proofs: Using induction on the rank of elements to derive formulas for the probability of commutativity and associativity.

Key Contributions and Results

• Commutativity Degree Approaching 1:
  The paper constructs a sequence of non-commutative, di-associative loops where the probability that a random commutator is trivial approaches 1 as the fold number $n$ increases. Specifically, for the central product $L = D_1 * D_2$ of two $n$-fold Cayley-Dickson loops over a finite center $Z$, the commutativity degree $P_c(L)$ is given by:
  $$P_c(L) = 1 - 6 \cdot \frac{1}{2^n} + 22 \cdot \frac{1}{4^n} - 24 \cdot \frac{1}{8^n} + 8 \cdot \frac{1}{16^n}$$
  As $n \to \infty$, $P_c(L) \to 1$. This contrasts sharply with non-abelian groups, where the commutativity degree is bounded above by $5/8$.

• Uniqueness of Central Product Decomposition:
  The authors prove that for $n \geq 3$, the decomposition of a central product of Cayley-Dickson loops is unique up to permutation. If $D_1 * \dots * D_m \cong E_1 * \dots * E_m$ where $D_i$ and $E_i$ are $n$-fold Cayley-Dickson loops, there exists a permutation $\sigma \in S_m$ such that $D_k \cong E_{\sigma(k)}$ for all $k$.

  • Mechanism: This result relies on the fact that for $n \geq 3$, the probability values associated with commutators of elements of a specific rank are strictly monotonic and distinct. An isomorphism must preserve these rank-based probabilities, thereby mapping constituent loops to constituent loops. This contrasts with the $n=2$ case (quaternion groups), where $Q_8 * Q_8 \cong (1, -1)_Z * (1, -1)_Z$ despite $Q_8 \not\cong (1, -1)_Z$.

• Associativity Degree:
  The paper calculates the associativity degree $P_A(L_n)$ for the $n$-fold Cayley-Dickson loop $L_n = (-1, \dots, -1)_Z$. This is defined as the probability that a random triple $(x, y, z)$ generates an associative subgroup. The derived formula is:
  $$P_A(L_n) = \frac{7 \cdot 4^n - 14 \cdot 2^n + 8}{8^n}$$
  The authors verify that for $n=2$, the degree is 1 (as expected for quaternion groups), and for $n=3$, the degree is $43/64$, matching previous results for Moufang loops.

Significance and Claims
The paper claims to demonstrate the "versatility" of di-associative loops compared to groups, particularly regarding their commutativity degrees. It establishes that Cayley-Dickson loops can be "close" to being groups in terms of commutativity (approaching 100% trivial commutators) while remaining non-associative.

Furthermore, the work provides a loop-theoretic analogue to results concerning tensor products of octonion algebras. By proving the uniqueness of decomposition for $n \geq 3$, the paper clarifies the structural rigidity of higher-fold Cayley-Dickson loops, distinguishing them from the more flexible $n=2$ case. The results serve as a foundation for understanding automorphisms and isomorphisms within these specific non-associative algebraic structures.