A Gröbner--Shirshov Basis for Nilpotent Rota--Baxter Algebras of Weight Zero
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910210824601600 |
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| author | Alhussein, H. |
| author_facet | Alhussein, H. |
| contents | We construct an explicit Gröbner--Shirshov basis for free associative Rota--Baxter algebras of weight zero with nilpotent operator $R^n=0$, where $n\ge 2$. First, we define a monomial order on the standard linear basis $RS(X)$ of the free algebra $R\mathrm{As}\langle X\rangle$ and establish fundamental identities for Rota--Baxter operators. For the case $n=2$, the basis consists of the Rota--Baxter relation $R(u)R(v)\to R(uR(v))+R(R(u)v)$ and the nilpotency relation $R(R(w))\to 0$. For general $n\ge 3$, we prove that the Gröbner--Shirshov basis is finite and consists of six families of relations $(R1)$--$(R6)$ derived from resolving all composition ambiguities. Using the Composition-Diamond Lemma, we describe the corresponding irreducible basis $\operatorname{Irr}(S)$, which provides normal forms for elements in the quotient algebra. This result gives a complete solution to the word problem for nilpotent Rota--Baxter algebras and establishes their operadic Gröbner--Shirshov basis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_09309 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Gröbner--Shirshov Basis for Nilpotent Rota--Baxter Algebras of Weight Zero Alhussein, H. Rings and Algebras We construct an explicit Gröbner--Shirshov basis for free associative Rota--Baxter algebras of weight zero with nilpotent operator $R^n=0$, where $n\ge 2$. First, we define a monomial order on the standard linear basis $RS(X)$ of the free algebra $R\mathrm{As}\langle X\rangle$ and establish fundamental identities for Rota--Baxter operators. For the case $n=2$, the basis consists of the Rota--Baxter relation $R(u)R(v)\to R(uR(v))+R(R(u)v)$ and the nilpotency relation $R(R(w))\to 0$. For general $n\ge 3$, we prove that the Gröbner--Shirshov basis is finite and consists of six families of relations $(R1)$--$(R6)$ derived from resolving all composition ambiguities. Using the Composition-Diamond Lemma, we describe the corresponding irreducible basis $\operatorname{Irr}(S)$, which provides normal forms for elements in the quotient algebra. This result gives a complete solution to the word problem for nilpotent Rota--Baxter algebras and establishes their operadic Gröbner--Shirshov basis. |
| title | A Gröbner--Shirshov Basis for Nilpotent Rota--Baxter Algebras of Weight Zero |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2605.09309 |