Rota-Baxter operators of nonzero weight on the split Cayley-Dickson algebra

Technical Summary: Rota-Baxter Operators of Nonzero Weight on the Split Octonions

Problem Statement
The paper addresses the classification of Rota-Baxter operators of nonzero weight on composition algebras. While Rota-Baxter operators of zero weight and nonzero weight on non-split composition algebras (division algebras) and lower-dimensional split algebras (direct sums of fields and matrix algebras $M_2(F)$) have been previously described, the case of split octonions (the split Cayley-Dickson algebra of dimension 8) remained open. Specifically, the author aims to describe all Rota-Baxter operators on split octonions over a field $F$ of characteristic $\neq 2$, distinguishing between "splitting" operators (those arising from a direct sum decomposition of the algebra into two subalgebras) and "non-splitting" operators.

Methodology
The author employs a structural approach based on the properties of the kernel and image of the Rota-Baxter operator $R$.

1. Definitions and Preliminaries: The paper utilizes the definition of a Rota-Baxter operator of weight $\lambda$: $R(x)R(y) = R(R(x)y + xR(y) + \lambda xy)$. A key distinction is made between splitting operators (where $R^2 = -\lambda R$) and non-splitting ones. The study focuses on the split Cayley-Dickson algebra $O$, defined as $M_2(F) + vM_2(F)$ with specific multiplication rules involving a symplectic involution.
2. Kernel Dimension Analysis: The author proves that for non-splitting operators, the dimension of the kernel must be either 3 or 4. This relies on the fact that the kernel and image are subalgebras, and the non-existence of 7-dimensional subalgebras in octonions (citing Knarr and Stroppel [7]).
3. Case Analysis by Kernel Dimension:
   • Four-dimensional Kernel: The author demonstrates that any non-splitting operator with a 4-dimensional kernel is equivalent, up to automorphism, anti-automorphism, and scalar multiplication, to a specific known operator $R_1$.
   • Three-dimensional Kernel: The paper investigates 3-dimensional kernels. It first rules out kernels containing idempotents (proving such operators are splitting). It then constructs a specific non-splitting operator $R_2$ with a kernel spanned by $\{e_{12}, ve_{12}, ve_{22}\}$ and proves that any non-splitting operator with such a kernel is equivalent to $R_2$.
4. Transformation Equivalence: The author introduces a transformation $\phi(R) = -R - \lambda \text{id}$, showing that $\phi(R_2)$ is equivalent to $R_1$. This unifies the classification of non-splitting operators.
5. Decomposition into Subalgebras: To classify splitting operators, the paper enumerates all possible decompositions of the split octonions into a direct sum of two subalgebras ($O = A_1 \oplus A_2$). This is performed over a quadratically closed field of characteristic $\neq 2$. The analysis covers cases where the subalgebras are unital or non-unital and of various dimensions (4, 5, 6).

Key Contributions and Results

1. Classification of Non-Splitting Operators: The paper establishes that, up to conjugation by automorphisms/anti-automorphisms, scalar multiplication, and the action of $\phi$, there exists exactly one non-splitting Rota-Baxter operator of nonzero weight on the split octonions over any field. This operator is represented by $R_1$, which acts as $-\text{id}$ on a specific 4-dimensional subalgebra $B_1$ and $0$ on a complementary 4-dimensional subspace $B_2$, with a specific mapping $R_1(e_{22}) = e_{11}$.
2. Classification of Splitting Operators: The author provides a complete list of all decompositions of split octonions into a sum of two subalgebras over a quadratically closed field. There are exactly seven non-isomorphic decompositions (Theorem 2), covering various dimensional combinations (e.g., $6+2$, $5+3$, $4+4$). These decompositions correspond to the splitting Rota-Baxter operators.
3. Completeness of Classification: By combining the results for non-splitting operators and the enumeration of splitting operators (decompositions), the paper claims to complete the classification of Rota-Baxter operators of nonzero weight on all composition algebras.

Significance
The paper concludes that the problem of classifying Rota-Baxter operators on composition algebras is now fully resolved. It extends previous work on simple algebras and superalgebras (such as $M_2(F)$ and $M_3(\mathbb{C})$) to the most complex case of composition algebras: the split octonions. The results rely heavily on the structural properties of subalgebras within the octonions, specifically utilizing the classification of subalgebras up to automorphism established by Knarr and Stroppel. The work provides a definitive answer to the structure of these operators, confirming that non-splitting operators are unique up to the specified transformations, while splitting operators are in one-to-one correspondence with the seven identified algebraic decompositions.