arXiv subjects

Dilei Lu

Publications and source records attributed to Dilei Lu.

6 recordsLinked to original sources

arXiv2026

Quasi-triangular Jordan D-bialgebras and extended relative Rota-Baxter operators

This paper introduces quasi-triangular Jordan D-bialgebras, which are constructed from solutions of the Jordan Yang-Baxter equation (JYBE) with invariant symmetric parts. We first develop a systematic operator approach to the JYBE by introducing the notion of extended relative Rota-Baxter operators on Jordan algebras. Such operators with their extensions are shown to induce new Jordan algebra structures. Moreover, the symmetrizer-antisymmetrizer decomposition of linear maps reduces the study of extended relative Rota-Baxter operators to both pairs of homomorphisms of Jordan algebras and relative Rota-Baxter operators, respectively. This operator framework subsequently yields a characterization of solutions of the JYBE whose symmetric parts are invariant. These operator forms are further investigated in the context of quadratic Jordan algebras and semi-direct product Jordan algebras, respectively, leading to explicit constructions of solutions of the JYBE. A factorizable Jordan D-bialgebra is then introduced as a special class of quasi-triangular ones, which naturally factorizes the underlying Jordan algebra. We prove that the Drinfeld classical double of any Jordan D-bialgebra admits a factorizable Jordan D-bialgebra structure. Finally, the operator perspective gives rise to the notion of quadratic Rota-Baxter Jordan algebras: those of zero weight yield triangular Jordan D-bialgebras, while those of nonzero weight are shown to be in one-to-one correspondence with factorizable Jordan D-bialgebras.

math.RT↗

arXiv2026

Post-Poisson algebras, extended $\mathcal{O}$-operators and extended Poisson Yang-Baxter equations

This paper introduces the extended $\mathcal{O}$-operators on Poisson algebras as a natural generalization of ordinary $\mathcal{O}$-operators, together with the extended Poisson Yang-Baxter equations. We show that $\mathcal{O}$-operators of weight $\lambda$ on Poisson algebras give rise to post-Poisson algebras, whose operad are the trisuccessor of the operad of Poisson algebras, and that extended $\mathcal{O}$-operators induce new Poisson algebra structures on module spaces.Equivalent characterizations of extended $\mathcal{O}$-operators are obtained via the symmetrizer-antisymmetrizer decomposition. The generalized Poisson Yang-Baxter equations are also introduced, and their connections with coboundary Poisson bialgebras and extended $\mathcal{O}$-operators are established. The tensor form of extended $\mathcal{O}$-operators leads to the notion of the extended Poisson Yang-Baxter equations, which generalizes the notion of the Poisson Yang-Baxter equations. Finally, the relationships among extended $\mathcal{O}$-operators, the extended Poisson Yang-Baxter equations, and the Poisson Yang-Baxter equations are studied in the framework of quadratic Poisson algebras and semi-direct product Poisson algebras.

math.RA↗

arXiv2026

Skew-symmetric Frobenius dialgebras, diassociative bialgebras and Yang-Baxter equations

A diassociative algebra (or dialgebra), a generalization of associative algebras with two associative products, is a fundamental algebraic structure of great importance and wide-ranging applications. In this paper, we establish a bialgebra theory for dialgebras completely and systematically. Explicitly, we introduce the notion of diassociative bialgebras (bi-dialgebras) which are equivalent to double constructions of Frobenius dialgebras as well as matched pairs of dialgebras. Moreover, we show that a bi-dialgebra gives rise to a Leibniz bialgebra, thereby lifting the classical relation between dialgebras and Leibniz algebras to the bialgebra level. In the coboundary cases, we introduce the diassociative Yang-Baxter equation (DAYBE) in a dialgebra, and solutions whose skew-symmetric part is invariant give rise to the so-called quasi-triangular bi-dialgebras. Moreover, weighted $\mathcal{O}$-operators on dialgebras are defined, which afford an operator interpretation for such solutions of the DAYBE. Finally, we also develop the triangular and factorizable theories for bi-dialgebras, and introduce skew-symmetric Rota-Baxter Frobenius dialgebras of arbitrary weight. We show that skew-symmetric Rota-Baxter Frobenius dialgebras of weight zero give rise to triangular bi-dialgebras, whereas those of nonzero weight induce factorizable bi-dialgebras. As an application, we provide concrete examples of factorizable bi-dialgebras from associative algebras and dialgebras, respectively.

math.RA↗

arXiv2026

Properties, deformation quantizations and bialgebras for dual pre-Poisson algebras

A dual pre-Poisson algebra is an algebraic structure that integrates a permutative algebra and a Leibniz algebra under certain compatibility conditions. As the Koszul dual notion of the pre-Poisson algebra, this structure serves as a natural generalization of the Poisson algebra. In this paper, we commence a study on dual pre-Poisson algebras from the algebraic point of view and establish a bialgebra theory for dual pre-Poisson algebras. We begin by investigating the fundamental properties of dual pre-Poisson algebras and provide several explicit constructions. In particular, we prove that the operad of dual pre-Poisson algebras is Koszul, and we compute its Hilbert-Poincar\'e series and codimension. Furthermore, we introduce the notion of diassociative formal deformations of permutative algebras and show that dual pre-Poisson algebras are the corresponding semi-classical limits. Moreover, we introduce dual pre-Poisson bialgebras, which are characterized both by Manin triples and by matched pairs of dual pre-Poisson algebras, thus developing a bialgebra theory for dual pre-Poisson algebras. Then our study leads to the permutative-Leibniz Yang-Baxter equation (PLYBE) that is composed of the permutative and the classical Leibniz Yang-Baxter equation. We conclude by introducing $\mathcal{O}$-operators and pre-dual pre-Poisson algebras, which provide a systematic method for constructing symmetric solutions to the PLYBE and, consequently, dual pre-Poisson bialgebras.

math.RA↗

arXiv2025

Quasi-triangular and factorizable Poisson bialgebras

In this paper, we introduce the notions of quasi-triangular and factorizable Poisson bialgebras. A factorizable Poisson bialgebra induces a factorization of the underlying Poisson algebra. We prove that the Drinfeld classical double of a Poisson bialgebra naturally admits a factorizable Poisson bialgebra structure. Furthermore, we introduce the notion of quadratic Rota-Baxter Poisson algebras and show that a quadratic Rota-Baxter Poisson algebra of zero weight induces a triangular Poisson bialgebra. Moreover, we establish a one-to-one correspondence between factorizable Poisson bialgebras and quadratic Rota-Baxter Poisson algebras of nonzero weights. Finally, we establish the quasi-triangular and factorizable theories for differential antisymmetric infinitesimal (ASI) bialgebras, and construct quasi-triangular and factorizable Poisson bialgebras from quasi-triangular and factorizable (commutative and cocommutative) differential ASI bialgebras respectively.

math.RA↗

arXiv2025

A bialgebra theory of post-Lie algebras via Manin triples and generalized Hessian Lie groups

We develop a bialgebra theory of post-Lie algebras that can be characterized by Manin triples of post-Lie algebras associated to a bilinear form satisfying certain invariant conditions. In the absence of dual representations for adjoint representations of post-Lie algebras, we utilize the geometric interpretation of post-Lie algebras to find the desired invariant condition, by generalizing pseudo-Hessian Lie groups to allow constant torsion for the flat connection. The resulting notion is a generalized pseudo-Hessian post-Lie algebra, which is a post-Lie algebra equipped with a nondegenerate symmetric invariant bilinear form. Moreover, generalized pseudo-Hessian post-Lie algebras are also naturally obtained from quadratic Rota-Baxter Lie algebras of weight one. On the other hand, the notion of partial-pre-post-Lie algebra (pp-post-Lie algebras) is introduced as the algebraic structure underlying generalized pseudo-Hessian post-Lie algebras, by splitting one of the two binary operations of post-Lie algebras. The notion of pp-post-Lie bialgebras is introduced as the equivalent structure of Manin triples of post-Lie algebras associated to a nondegenerate symmetric invariant bilinear form, thereby establishing a bialgebra theory for post-Lie algebras via the Manin triple approach. We also study the related analogs of the classical Yang-Baxter equation, $\mathcal O$-operators and successors for pp-post-Lie algebras. In particular, there is a construction of pp-post-Lie bialgebras from the successors of pp-post-Lie algebras.

math.QA↗