A further study on averaging commutative and cocommutative infinitesimal bialgebras and special apre-perm bialgebras
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2025
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| author | Zhao, Quan Liu, Guilai |
| author_facet | Zhao, Quan Liu, Guilai |
| contents | In order to generalize the fact that an averaging commutative algebra gives rise to a perm algebra to the bialgebra level, the notion of a special apre-perm algebra was introduced as a new splitting of perm algebras, and it has been shown that an averaging commutative and cocommutative infinitesimal bialgebra gives rise to a special apre-perm bialgebra. In this paper, we give a further study on averaging commutative and cocommutative infinitesimal bialgebras and special apre-perm bialgebras. A solution of the averaging associative Yang-Baxter equation whose symmetric part is invariant gives rise to an averaging commutative and cocommutative infinitesimal bialgebra that is called quasi-triangular, and such solutions can be equivalently characterized as $\mathcal{O}$-operators of admissible averaging commutative algebras with weights. Moreover assuming the symmetric parts of such solutions to be zero or nondegenerate, we obtain typical subclasses of quasi-triangular averaging commutative and cocommutative infinitesimal bialgebras, namely the triangular and factorizable ones respectively. Both of them are shown to closely relate to symmetric averaging Rota-Baxter Frobenius commutative algebras. There is a parallel procedure developed for special apre-perm bialgebras. In particular, the fact that an averaging commutative and cocommutative infinitesimal bialgebra gives rise to a special apre-perm bialgebra is still available when these bialgebras are limited to the quasi-triangular cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_09208 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A further study on averaging commutative and cocommutative infinitesimal bialgebras and special apre-perm bialgebras Zhao, Quan Liu, Guilai Rings and Algebras 17A36, 17A40, 17B10, 17B38, 17D25, 18M70 In order to generalize the fact that an averaging commutative algebra gives rise to a perm algebra to the bialgebra level, the notion of a special apre-perm algebra was introduced as a new splitting of perm algebras, and it has been shown that an averaging commutative and cocommutative infinitesimal bialgebra gives rise to a special apre-perm bialgebra. In this paper, we give a further study on averaging commutative and cocommutative infinitesimal bialgebras and special apre-perm bialgebras. A solution of the averaging associative Yang-Baxter equation whose symmetric part is invariant gives rise to an averaging commutative and cocommutative infinitesimal bialgebra that is called quasi-triangular, and such solutions can be equivalently characterized as $\mathcal{O}$-operators of admissible averaging commutative algebras with weights. Moreover assuming the symmetric parts of such solutions to be zero or nondegenerate, we obtain typical subclasses of quasi-triangular averaging commutative and cocommutative infinitesimal bialgebras, namely the triangular and factorizable ones respectively. Both of them are shown to closely relate to symmetric averaging Rota-Baxter Frobenius commutative algebras. There is a parallel procedure developed for special apre-perm bialgebras. In particular, the fact that an averaging commutative and cocommutative infinitesimal bialgebra gives rise to a special apre-perm bialgebra is still available when these bialgebras are limited to the quasi-triangular cases. |
| title | A further study on averaging commutative and cocommutative infinitesimal bialgebras and special apre-perm bialgebras |
| topic | Rings and Algebras 17A36, 17A40, 17B10, 17B38, 17D25, 18M70 |
| url | https://arxiv.org/abs/2510.09208 |