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Algebras constructed by Rota-Baxter operators

A. S. Dzhumadil'daev

TL;DR

This work analyzes RB-algebras $AR=(A,\circ)$ built from an associative commutative base algebra $A$ with a Baxter operator $R$ of nonzero weight $\lambda$, proving that the basic identities $rcom$ and $f_4$ govern degree-$3$ and degree-$4$ multilinear identities and that, for $AR^-$ (Lie) and $AR^+$ (Jordan), Tortkara and Tortkara-type structures arise. It further shows that all degree-$4$ identities for $AR^-$ follow from Tortkara, while degree-$5$ identities for $AR^+$ are controlled by a corresponding $f_5$-type condition, with $f_4'$ not implying $f_5^+$. The paper also explores various explicit RB-multiplications, including $m_\varepsilon$, Zinbiel- and Novikov-related products, and a family of star-products $\star_{k,n}$, establishing associativity, Zinbiel, and right-commutative properties in these families. Additional constructions include Lie brackets $[a,b]_n$ on Zinbiel products, showing Tortkara identities, and several Zinbiel/RB-algebra instances derived from geometric-type operators such as integrals, highlighting the richness of RB-algebras in both theoretical and constructive directions.

Abstract

For associative commutative algebras $A$ with Rota-Baxter operator $R$ identities of the algebra $AR=(A,\circ)$, where $a\circ b= aR(b),$ are found.

Algebras constructed by Rota-Baxter operators

TL;DR

This work analyzes RB-algebras built from an associative commutative base algebra with a Baxter operator of nonzero weight , proving that the basic identities and govern degree- and degree- multilinear identities and that, for (Lie) and (Jordan), Tortkara and Tortkara-type structures arise. It further shows that all degree- identities for follow from Tortkara, while degree- identities for are controlled by a corresponding -type condition, with not implying . The paper also explores various explicit RB-multiplications, including , Zinbiel- and Novikov-related products, and a family of star-products , establishing associativity, Zinbiel, and right-commutative properties in these families. Additional constructions include Lie brackets on Zinbiel products, showing Tortkara identities, and several Zinbiel/RB-algebra instances derived from geometric-type operators such as integrals, highlighting the richness of RB-algebras in both theoretical and constructive directions.

Abstract

For associative commutative algebras with Rota-Baxter operator identities of the algebra , where are found.
Paper Structure (16 sections, 6 theorems, 610 equations)

This paper contains 16 sections, 6 theorems, 610 equations.

Table of Contents

  1. Introduction
  2. Identities of RB-algebras of degree 3 and 4
  3. Identities of RB-algebras of degree 5
  4. Multi-linear identities of $AR^-$ in degree $4$
  5. Multilinear right-commutative identity in degree 4
  6. Multi-linear identities of $AR^+$ in degree $5$
  7. Algebras with multiplication $m_\varepsilon$
  8. An algebra with multiplication $e_i\circ e_j=\varepsilon_{i,j} e_i,$ where $\varepsilon_{i,j}=0,$ if $i\le j$ and $\varepsilon_{i,j}=\varepsilon_j,$ if $i>j.$
  9. The multiplication $\star_{k,n}$
  10. Lie commutators of Zinbiel products
  11. Multiplications with identity $f_4'(a,b,c,d)=0$
  12. The multiplication $a\star_{0,n} b=\sum_{i=0}^n {n\choose i}\int_i a \int_{n-i} b$ is associative
  13. $AR$-algebra of Novikov algebra
  14. $AR$-algebra of Zinbiel algebra
  15. The multiplication $a\star b=\int a \int b$
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $A$ be associative commutative algebra over a field of characteristic $0$ and $R:A\rightarrow A$ is Rota-Baxter operator with non-zero weight $\lambda.$ Then the algebra $AR=(A,\circ),$ where $a\circ b=a\,R(b),$ satisfies the following identities Any identity of the algebra $AR$ of degree no more than $4$ are consequences of the identities $rcom=0$ and $f_4=0.$ The algebra $AR$ under Lie comm

Theorems & Definitions (6)

  • Theorem 1.1
  • Corollary 2.1
  • Theorem 10.1
  • Proposition 13.1
  • Proposition 13.2
  • Proposition 13.3