Quasi-triangular and factorizable perm bialgebras
Yuanchang Lin
TL;DR
The paper develops quasi-triangular and factorizable theories for perm bialgebras by extending the perm Yang-Baxter framework. It defines quasi-triangular structures via $r\in A\otimes A$ solving the perm YBE with an $(R,\mathrm{ad})$-invariant skew-symmetric part, and introduces factorizable cases when $T_{r-\sigma(r)}$ is nondegenerate; it further builds a bridge to Rota-Baxter theory through relative Rota-Baxter operators and quadratic Rota-Baxter perm algebras. A key result is a one-to-one correspondence between factorizable perm bialgebras and quadratic Rota-Baxter perm algebras of nonzero weight, with weight zero giving triangular structures; the double perm algebra provides a canonical factorizable structure. Together, these developments extend classical quasi-triangular/factorizable bialgebra concepts to perm algebras and link them to operator-theoretic frameworks, with potential implications for integrable systems and combinatorial algebra.
Abstract
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.