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Huhu Zhang

Publications and source records attributed to Huhu Zhang.

14 recordsLinked to original sources
arXiv2026

Disuccessors, Gr\"obner-Shirshov bases and free L-dendriform algebras

In this paper, we prove respectively that the disuccessor operations on the associative operad $\as$, the Lie operad $\lie$, and the pre-Lie operad $\prelie$ preserve Gr\"obner-Shirshov bases. This structural preservation enables the transfer of known bases to more complex operads. As a consequence, we introduce new methods for constructing Gr\"obner-Shirshov bases for the $\dend$ and $\prelie$ operads, and explicitly construct a Gr\"obner-Shirshov basis for the free L-dendriform algebra. This provides a conceptual and computationally efficient resolution of Madariaga's problem and offers new insights into the combinatorial structure of L-dendriform algebras.

math.RA↗
arXiv2026

Induced structures of operated algebras with applications to multi-Novikov algebras

We provide a general notion of induced structures of operated algebras in the context of unary-binary operads. This notion fully captures the binary quadratic relations encoded by a unary-binary operad, thereby unifying and formalizing the various constructions that have appeared in the literature under the informal term of ``induced structures''. As an application, we show that the Novikov algebra and the recently introduced multi-Novikov algebra are the induced structures of the differential commutative algebra and the multi-differential commutative algebra respectively. We also explicitly determine the induced structure of the noncommuting multi-differential commutative algebra.

math.CT↗
arXiv2026

General multi-Novikov algebras, multi-differential algebras and their free constructions

Motivated by the recent development of noncommutative Novikov algebras and multi-Novikov algebras from the study of regularity structures of stochastic PDEs, this paper gives a general approach to study various multi-Novikov algebras and multi-differential algebras, with close connection with Poisson algebras. The construction of S. Gelfand of Novikov algebras from differential commutative algebras is generalized to this context. Free noncommuting multi-Novikov algebras are constructed from typed decorated rooted trees and from noncommuting multi-differential polynomials with populated conditions.

math.RA↗
arXiv2025

Integro-differential rings on species and derived structures

In the theory of species, differential as well as integral operators are known to arise in a natural way. In this paper, we shall prove that they precisely fit together in the algebraic framework of integro-differential rings, which are themselves an abstraction of classical calculus (incorporating its Fundamental Theorem). The results comprise (set) species as well as linear species. Localization of (set) species leads to the more general structure of modified integro-differential rings, previously employed in the algebraic treatment of Volterra integral equations. Furthermore, the ring homomorphism from species to power series via taking generating series is shown to be a (modified) integro-differential ring homomorphism. As an application, a topology and further algebraic operations are imported to virtual species from the general theory of integro-differential rings.

math.CO↗
arXiv2024

Averaging operators on groups and Hopf algebras

Rota-Baxter operators on groups were studied quite recently. Motivated mainly by the fact that weight zero Rota-Baxter operators and averaging operators are Koszul dual to each other, we propose the concepts of averaging group and averaging Hopf algebra, and study relationships among them and the existing averaging Lie algebras. We also show that an averaging group induces a disemigroup and a rack, respectively. As the free object is one of the most significant objects in a category, we also construct explicitly the free averaging group on a set.

math.RA↗
arXiv2024

Weighted differential ($q$-tri)dendriform algebras

In this paper, we first introduce a weighted derivation on algebras over an operad $\cal P$, and prove that for the free $\cal P$-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential ($q$-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential ($q$-tri) dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts.

math.RA↗
arXiv2024

Some new operated Lie polynomial identities and Gr\"obner-Shirshov bases

Bremner and Elgendy developed a classification of operated polynomial identities for linear operators on associative algebras, encompassing both classical and newly discovered cases. Within the framework of Rota's Program, each of these new operated associative polynomial identities was shown to be Gr\"obner-Shirshov. This naturally led to a question posed by Guo and collaborators: is each corresponding operated Lie polynomial identity also Gr\"obner-Shirshov? In this paper, we provide an affirmative answer by proving that each such Lie analogue indeed is Gr\"obner-Shirshov, thereby enriching the development of Rota's Program on algebraic operators within the Lie algebraic setting.

math.RA↗
arXiv2023

Compatible structures of operads by polarization, their Koszul duality and Manin products

Algebraic structures with multiple copies of a given type of operations interrelated by various compatibility conditions have long being studied in mathematics and mathematical physics. They are broadly referred as linearly compatible, matching, and totally compatible structures. This paper gives a unified approach to these structures in the context of operads. We first generalize the process of polarization for polynomials in invariant theory to the one for operads, leading to the general notion of linearly compatible operads. Refining the polarization by partitioning it into foliations, we obtain a notion of matching operads consolidating those appeared recently from applications of regularity structures, and Volterra integral equations. Distinguished among them is the leveled matching compatibility, which is unique with respect to a fix ordering of the vertices in tree monomials. Equating all matching compatibilities of a given operad leads to the totally compatible operad of this operad. For unary/binary quadratic operads, the linear compatibility and the total compatibility are in Koszul dual to each other, and there is a Koszul self-duality among the matching compatibilities. In particular, the leveled matching compatibility is Koszul self-dual. For binary quadratic operads, these three compatible operads can also be obtained by taking Manin black and white products. For some finitely generated binary quadratic operad, Koszulity is preserved under taking the compatibilities.

math.CT↗
arXiv2023

Free weighted differential ($q$-tri)dendriform algebras

In the present paper, we propose the concepts of weighted differential ($q$-tri)dendriform algebras and give some basic properties of them. The corresponding free objects are constructed, in both the commutative and noncommutative contexts.

math.RA↗
arXiv2022

Free $Ω$-Rota-Baxter systems and Gröbner-Shirshov bases

In this paper, we propose the concept of an $Ω$-Rota-Baxter system, which is a generalization of a Rota-Baxter system and an $Ω$-Rota-Baxter algebra of weight zero. In the framework of operated algebras, we obtain a linear basis of a free $Ω$-Rota-Baxter system for an extended diassociative semigroup $Ω$, in terms of bracketed words and the method of Gröbner-Shirshov bases. As applications, we introduce the concepts of Rota-Baxter system family algebras and matching Rota-Baxter systems as special cases of $Ω$-Rota-Baxter systems, and construct their free objects. Meanwhile, free $Ω$-Rota-Baxter algebras of weight zero, free Rota-Baxter systems, free Rota-Baxter family algebras and free matching Rota-Baxter algebras are reconstructed via new method.

math.RA↗
arXiv2022

Operator identities on Lie algebras, rewriting systems and Gröbner-Shirshov bases

Motivated by the pivotal role played by linear operators, many years ago Rota proposed to determine algebraic operator identities satisfied by linear operators on associative algebras, later called Rota's program on algebraic operators. Recent progresses on this program have been achieved in the contexts of operated algebra, rewriting systems and Groebner-Shirshov bases. These developments also suggest that Rota's insight can be applied to determine operator identities on Lie algebras, and thus to put the various linear operators on Lie algebras in a uniform perspective. This paper carries out this approach, utilizing operated polynomial Lie algebras spanned by non-associative Lyndon-Shirshov bracketed words. The Lie algebra analog of Rota's program was formulated in terms convergent rewriting systems and equivalently in terms of Groebner-Shirshov bases. This Lie algebra analog is shown to be compatible with Rota's program for associative algebras. As applications, a classification of differential type operators and Rota-Baxter operators are presented.

math.QA↗
arXiv2021

Rota's program on algebraic operators, rewriting systems and Gröbner-Shirshov bases

Many years ago, Rota proposed a program on determining algebraic identities that can be satisfied by linear operators. After an extended period of dormant, progress on this program picked up speed in recent years, thanks to perspectives from operated algebras and Gröbner-Shirshov bases. These advances were achieved in a series of papers from special cases to more general situations. These perspectives also indicate that Rota's insight can be manifested very broadly, for other algebraic structures such as Lie algebras, and further in the context of operads. This paper gives a survey on the motivation, early developments and recent advances on Rota's program, for linear operators on associative algebras and Lie algebras. Emphasis will be given to the applications of rewriting systems and Gröbner-Shirshov bases. Problems, old and new, are proposed throughout the paper to prompt further developments on Rota's program.

math.RA↗
arXiv2021

Compatible structures on unary binary nonsymmetric operads with quadratic and cubic relations

Various compatibility conditions among replicated copies of operations in a given algebraic structure have appeared in broad contexts in recent years. Taking an uniform approach, this paper gives an operadic study of compatibility conditions for nonsymmetric operads with unary and binary operations, and homogeneous quadratic and cubic relations. This generalizes the previous studies for binary quadratic operads. We consider three compatibility conditions, namely the linear compatibility, matching compatibility and total compatibility, with increasingly strict restraints among the replicated copies. The linear compatibility is in Koszul dual to the total compatibility, while the matching compatibility is self dual. Further, each compatibility can be expressed in terms of either one or both of the two Manin square products.

math.CT↗