$O$-operators on associative algebras and associative Yang-Baxter equations

We introduce the concept of an extended O-operator that generalizes the well-known concept of a Rota-Baxter operator. We study the associative products coming from these operators and establish the relationship between extended O-operators and the associative Yang-Baxter equation, extended associative Yang-Baxter equation and generalized Yang-Baxter equation.

💡 Research Summary

The paper introduces a new algebraic concept called an “extended O‑operator,” which generalizes both classical O‑operators and Rota‑Baxter operators on associative algebras. Let A be an associative algebra over a field k, and let M be an A‑bimodule. A linear map T : M → A together with a bilinear map β : M × M → A is called an extended O‑operator of weight λ∈k if for all u, v∈M the following identity holds:
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