Extended Rota-Baxter algebras, diagonally colored Delannoy paths and Hopf algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929223039451136 |
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| author | Zheng, Shanghua Guo, Li Qiu, Huizhen |
| author_facet | Zheng, Shanghua Guo, Li Qiu, Huizhen |
| contents | The Rota-Baxter operator and the modified Rota-Baxter operator on various algebras are both important in mathematics and mathematical physics. The former is originated from the integration-by-parts formula and probability with applications to the renormalization of quantum field theory and the classical Yang-Baxter equation. The latter originated from Hilbert transformations with applications to ergodic theory and the modified Yang-Baxter equation. Their merged form, called the extended Rota-Baxter operators, has also found interesting applications recently.
This paper presents a systematic study of the extended Rota-Baxter operator. We show that while extended Rota-Baxter operators have properties similar to Rota-Baxter operators; they provide a linear structure that unifies Rota-Baxter operators and modified Rota-Baxter operators. Examples of extended Rota-Baxter operators are also given, especially from polynomials and Laurent series due to their importance in ($q$-)integration and the renormalization in quantum field theory. We then construct free commutative extended Rota-Baxter operators by a generalization of the quasi-shuffle product. The multiplication of the initial object in the category of commutative extended Rota-Baxter operators allows a combinatorial interpretation in terms of a color-enrichment of Delannoy paths. Applying its universal property, we equip a free commutative extended Rota-Baxter operators with a coproduct which has a cocycle condition, yielding a bialgebraic structure. We then show that this bialgebra on a free extended Rota-Baxter operators possesses an increasing filtration and a connectedness property, culminating at a Hopf algebraic structure on a free commutative extended Rota-Baxter operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11363 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extended Rota-Baxter algebras, diagonally colored Delannoy paths and Hopf algebras Zheng, Shanghua Guo, Li Qiu, Huizhen Rings and Algebras Mathematical Physics Combinatorics 16W99, 17B37, 16S10, 16T10, 16T30, 57R56 The Rota-Baxter operator and the modified Rota-Baxter operator on various algebras are both important in mathematics and mathematical physics. The former is originated from the integration-by-parts formula and probability with applications to the renormalization of quantum field theory and the classical Yang-Baxter equation. The latter originated from Hilbert transformations with applications to ergodic theory and the modified Yang-Baxter equation. Their merged form, called the extended Rota-Baxter operators, has also found interesting applications recently. This paper presents a systematic study of the extended Rota-Baxter operator. We show that while extended Rota-Baxter operators have properties similar to Rota-Baxter operators; they provide a linear structure that unifies Rota-Baxter operators and modified Rota-Baxter operators. Examples of extended Rota-Baxter operators are also given, especially from polynomials and Laurent series due to their importance in ($q$-)integration and the renormalization in quantum field theory. We then construct free commutative extended Rota-Baxter operators by a generalization of the quasi-shuffle product. The multiplication of the initial object in the category of commutative extended Rota-Baxter operators allows a combinatorial interpretation in terms of a color-enrichment of Delannoy paths. Applying its universal property, we equip a free commutative extended Rota-Baxter operators with a coproduct which has a cocycle condition, yielding a bialgebraic structure. We then show that this bialgebra on a free extended Rota-Baxter operators possesses an increasing filtration and a connectedness property, culminating at a Hopf algebraic structure on a free commutative extended Rota-Baxter operator. |
| title | Extended Rota-Baxter algebras, diagonally colored Delannoy paths and Hopf algebras |
| topic | Rings and Algebras Mathematical Physics Combinatorics 16W99, 17B37, 16S10, 16T10, 16T30, 57R56 |
| url | https://arxiv.org/abs/2401.11363 |