Quasi-triangular and factorizable perm bialgebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915254914514944 |
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| author | Lin, Yuanchang |
| author_facet | Lin, Yuanchang |
| contents | In this paper, we introduce the notions of quasi-triangular and factorizable perm bialgebras, based on notions of the perm Yang-Baxter equation and $(R, \mathrm{ad})$-invariant condition. A factorizable perm bialgebra induces a factorization of the underlying perm algebra and the double of a perm bialgebra naturally admits a factorizable perm bialgebra structure. The notion of relative Rota-Baxter operators of weights on perm algebras is introduced to characterize solutions of the perm Yang-Baxter equation, whose skew-symmetric parts are $(R, \mathrm{ad})$-invariant. These operators are in one-to-one correspondence with linear transformations fulfilling a Rota-Baxter-type identity in the case of quadratic perm algebras. Furthermore, we introduce the notion of quadratic Rota-Baxter perm algebras of weights, demonstrate that a quadratic Rota-Baxter perm algebra of weight $0$ induces a triangular perm bialgebra, and establish a one-to-one correspondence between quadratic Rota-Baxter perm algebras of nonzero weights and factorizable perm bialgebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_16495 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-triangular and factorizable perm bialgebras Lin, Yuanchang Representation Theory 16T10, 16T25, 16W99, 17B38 In this paper, we introduce the notions of quasi-triangular and factorizable perm bialgebras, based on notions of the perm Yang-Baxter equation and $(R, \mathrm{ad})$-invariant condition. A factorizable perm bialgebra induces a factorization of the underlying perm algebra and the double of a perm bialgebra naturally admits a factorizable perm bialgebra structure. The notion of relative Rota-Baxter operators of weights on perm algebras is introduced to characterize solutions of the perm Yang-Baxter equation, whose skew-symmetric parts are $(R, \mathrm{ad})$-invariant. These operators are in one-to-one correspondence with linear transformations fulfilling a Rota-Baxter-type identity in the case of quadratic perm algebras. Furthermore, we introduce the notion of quadratic Rota-Baxter perm algebras of weights, demonstrate that a quadratic Rota-Baxter perm algebra of weight $0$ induces a triangular perm bialgebra, and establish a one-to-one correspondence between quadratic Rota-Baxter perm algebras of nonzero weights and factorizable perm bialgebras. |
| title | Quasi-triangular and factorizable perm bialgebras |
| topic | Representation Theory 16T10, 16T25, 16W99, 17B38 |
| url | https://arxiv.org/abs/2504.16495 |