[Paper Review] Classification of Rota-Baxter operators on semigroup algebras of order two and three

Li Guo, Markus Rosenkranz+1 more|arXiv (Cornell University)|Feb 15, 2014

Advanced Topics in Algebra1 references3 citations

TL;DR

This paper classifies all Rota-Baxter operators of weight zero on semigroup algebras of order two and three using computer algebra, specifically Mathematica, by solving the defining matrix equations systematically. It provides a complete list of such operators, distinguishing between commutative and noncommutative cases, and introduces a verified computational procedure for prediction and validation of solutions.

ABSTRACT

In this paper we determine all the Rota-Baxter operators of weight zero on semigroup algebras of order two and three with the help of computer algebra. We determine the matrices for these Rota-Baxter operators by directly solving the defining equations of the operators. We also produce a Mathematica procedure to predict and verify these solutions.

Motivation & Objective

• To systematically classify all Rota-Baxter operators of weight zero on semigroup algebras of order two and three.
• To provide explicit matrix representations of these operators using the canonical basis of semigroup algebras.
• To develop and validate a computer algebra procedure in Mathematica for predicting and verifying solutions.
• To distinguish between commutative and noncommutative semigroup algebras in the classification.
• To ensure theoretical rigor by combining computational predictions with formal algebraic proofs for each case.

Proposed method

• Formulate the defining equations for Rota-Baxter operators of weight zero in matrix form using structure constants derived from semigroup multiplication tables.
• Apply the Rota-Baxter identity $ P(x)P(y) = P(xP(y)) + P(P(x)y) $ to generate a system of polynomial equations in the matrix entries of $ P $.
• Use symbolic computation in Mathematica to solve the resulting system of equations, with functions like RBA (to generate equations) and FindRBO (to solve them).
• Convert semigroup Cayley tables into structure constants $ r^m_{kl} $ via the SGM function to parameterize the algebraic structure.
• Verify solutions by checking consistency with the Rota-Baxter identity and by comparing with known classifications in the literature.
• Handle cases by case analysis, distinguishing between zero and non-zero entries in matrix entries to enumerate all possible solutions.

Experimental results

Research questions

• RQ1 What are all the Rota-Baxter operators of weight zero on semigroup algebras of order two?
• RQ2 How do the Rota-Baxter operators on commutative semigroup algebras of order three differ from those on noncommutative ones?
• RQ3 Can a computer algebra system accurately predict and verify Rota-Baxter operator solutions for low-dimensional semigroup algebras?
• RQ4 What is the complete set of matrix solutions for Rota-Baxter operators on each isomorphism class of semigroup algebras of order three?
• RQ5 How do the solutions vary depending on the semigroup’s multiplication structure, particularly in terms of zero patterns and parameter dependencies?

Key findings

• All Rota-Baxter operators of weight zero on semigroup algebras of order two are classified, with solutions expressed as matrices parameterized by elements of the base ring.
• For commutative semigroup algebras of order three, 12 isomorphism classes were analyzed, and solutions were derived for each, including families like $ C_{1,1} $ with six free parameters.
• For noncommutative semigroup algebras of order three, six isomorphism classes were studied, yielding solutions such as $ N_{6,7} $, $ N_{6,8} $, and $ N_{6,9} $, parameterized by non-zero elements of the base ring.
• The solution $ N_{6,9} = \begin{pmatrix} \frac{ab}{a-b} & a & \frac{a^2}{b-a} \\ 0 & 0 & 0 \\ \frac{b^2}{a-b} & b & \frac{ab}{b-a} \end{pmatrix} $ was derived for $ \mathbf{k}[NCS(6)] $, valid for $ a, b \in \mathbf{k} \setminus \{0\} $, $ a \neq b $.
• The Mathematica procedure successfully predicted and verified all solutions, with outputs confirming known solutions and identifying special cases.
• The classification reveals that Rota-Baxter operators on semigroup algebras take significantly different forms compared to general associative algebras, highlighting the importance of the semigroup basis.