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Zuan Liu

Publications and source records attributed to Zuan Liu.

2 recordsLinked to original sources
arXiv2025

Polygraphic resolutions for operated algebras

This paper introduces the structure of operated polygraphs as a categorical model for rewriting in operated algebras, generalizing Gr\"obner-Shirshov bases with non-monomial termination orders. We provide a combinatorial description of critical branchings of operated polygraphs using the structure of polyautomata that we introduce in this paper. Polyautomata extend linear polygraphs equipped with an operator structure formalized by a pushdown automaton. We show how to construct polygraphic resolutions of free operated algebras from their confluent and terminating presentations. Finally, we apply our constructions to several families of operated algebras, including Rota-Baxter algebras, differential algebras, and differential Rota-Baxter algebras.

math.RA↗
arXiv2023

Gröbner-Shirshov bases and linear bases for free multi-operated algebras over algebras with applications to differential Rota-Baxter algebras and integro-differential algebras

Quite much recent studies has been attracted to the operated algebra since it unifies various notions such as the differential algebra and the Rota-Baxter algebra. An $Ω$-operated algebra is a an (associative) algebra equipped with a set $Ω$ of linear operators which might satisfy certain operator identities such as the Leibniz rule. A free $Ω$-operated algebra $B$ can be generated on an algebra $A$ similar to a free algebra generated on a set. If $A$ has a Gröbner-Shirshov basis $G$ and if the linear operators $Ω$ satisfy a set $Φ$ of operator identities, it is natural to ask when the union $G\cup Φ$ is a Gröbner-Shirshov basis of $B$. A previous work answers this question affirmatively under a mild condition, and thereby obtains a canonical linear basis of $B$. In this paper, we answer this question in the general case of multiple linear operators. As applications we get operated Gröbner-Shirshov bases for free differential Rota-Baxter algebras and free integro-differential algebras over algebras as well as their linear bases. One of the key technical difficulties is to introduce new monomial orders for the case of two operators, which might be of independent interest.

math.RA↗