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Jiafeng Lü

Publications and source records attributed to Jiafeng Lü.

15 recordsLinked to original sources
arXiv2026

Universal Enveloping Algebras of Hom-Lie Algebras with $α$-Derivations

In this paper, we construct nonunital universal enveloping Hom-associative algebras for multiplicative Hom-Lie algebras equipped with $α$-derivations. We prove that the prescribed derivation extends to the enveloping algebra and that the resulting universal property yields a left adjoint to the commutator functor between the corresponding categories. When the twisting map is bijective, we establish a canonical isomorphism between this enveloping algebra and the twist of the nonunital universal enveloping algebra of the associated untwisted Lie algebra. This isomorphism preserves the twisting maps and the derivations. Consequently, the classical Poincaré-Birkhoff-Witt theorem identifies the associated graded algebra for the untwisted product with the positive-degree symmetric algebra. When the Hom-Lie algebra is finite-dimensional, the resulting PBW basis consists of ordered monomials of positive length. The twisting map, the extended $α$-derivation, and its untwisted counterpart preserve the PBW filtration.

math.RA↗
arXiv2026

Nijenhuis operators on 2D pre-Lie algebras and 3D associative algebras

In this paper, we describe all Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras. As an application, using these operators, we obtain solutions of the classical Yang-Baxter equation on the corresponding sub-adjacent Lie algebras.

math-ph↗
arXiv2025

Manin triples, bialgebras and Yang-Baxter equation of $A_3$-associative algebras

$A_3$-associative algebra is a generalization of associative algebra and is one of the four remarkable types of Lie-admissible algebras, along with associative algebra, left-symmetric algebra and right-symmetric algebra. This paper develops bialgebra theory for $A_3$-associative algebras. We introduce Manin triples and bialgebras for $A_3$-associative algebras, prove their equivalence using matched pairs of $A_3$-associative algebras, and define the $A_3$-associative Yang-Baxter equation and triangular $A_3$-associative bialgebras. Additionally, we introduce relative Rota-Baxter operators to provide skew-symmetric solutions of the $A_3$-associative Yang-Baxter equation.

math.RA↗
arXiv2025

Matched pairs and double construction bialgebras of (transposed) Poisson 3-Lie algebras

Double construction bialgebras for Poisson 3-Lie algebras and transposed Poisson 3-Lie algebras are defined and studied using matched pairs. Poisson 3-Lie algebras and transposed Poisson 3-Lie algebras are constructed on direct sums and tensor products of vector spaces. Matched pairs, Manin triples, and double construction Poisson 3-Lie bialgebras are shown to be equivalent. Since the double construction approach does not apply to bialgebra theory for transposed Poisson 3-Lie algebras, admissible transposed Poisson 3-Lie algebras are introduced. These algebras are both transposed Poisson 3-Lie algebras and Poisson 3-Lie algebras. An equivalence between matched pairs and double construction admissible transposed Poisson 3-Lie bialgebras is established.

math.RA↗
arXiv2025

Universal enveloping H-pseudoalgebras of DGP pseudoalgebras

The notions of Poisson $H$-pseudoalgebras are generalizations of Poisson algebras in a pseudotensor category $\mathcal{M}^{\ast}(H)$. This paper introduces an analogue of Poisson-Ore extension in Poisson $H$-pseudoalgebras. Poisson $H$-pseudoalgebras with the differential graded setting induces the notions of differential graded Poisson $H$-pseudoalgebras (DGP pseudoalgebras, for short). The DGP pseudoalgebra with some compatibility conditions is proved to be closed under tensor product. Furthermore, the universal enveloping $H$-pseudoalgebras of DGP pseudoalgebras are constructed by a $\mathcal{P}$-triple. A unique differential graded pseudoalgebra homomorphism between a universal enveloping $H$-pseudoalgebra of a DGP pseudoalgebra and a $\mathcal{P}$-triple of a DGP pseudoalgebra is obtained.

math.AC↗
arXiv2024

Universal enveloping algebras of weighted differential Poisson algebras

The $λ$-differential operators and modified $λ$-differential operators are generalizations of classical differential operators. This paper introduces the notions of $λ$-differential Poisson ($λ$-DP for short) algebras and modified $λ$-differential Poisson ($λ$-mDP for short) algebras as generalizations of differential Poisson algebras. The $λ$-DP algebra is proved to be closed under tensor product, and a $λ$-DP algebra structure is provided on the cohomology algebra of the $λ$-DP algebra. These conclusions are also applied to $λ$-mDP algebras and their modules. Finally, the universal enveloping algebras of $λ$-DP algebras are generalized by constructing a $\mathcal{P}$-triple. Three isomorphisms among opposite algebras, tensor algebras and the universal enveloping algebras of $λ$-DP algebras are obtained.

math-ph↗
arXiv2024

Relative torsionfreeness and Frobenius extensions

Let $S/R$ be a Frobenius extension with $_RS_R$ centrally projective over $R$. We show that if $_Rω$ is a Wakamatsu tilting module then so is $_SS\otimes_Rω$, and the natural ring homomorphism from the endomorphism ring of $_Rω$ to the endomorphism ring of $_SS\otimes_Rω$ is a Frobenius extension in addition that pd$(ω_T)$ is finite, where $T$ is the endomorphism ring of $_Rω$. We also obtain that the relative $n$-torsionfreeness of modules is preserved under Frobenius extensions. Furthermore, we give an application, which shows that the generalized G-dimension with respect to a Wakamatsu module is invariant under Frobenius extensions.

math.RA↗
arXiv2024

Manin triples associated to $n$-Lie bialgebras

In this paper, we study the Manin triples associated to $n$-Lie bialgebras. We introduce the concept of operad matrices for $n$-Lie bialgebras. In particular, by studying a special case of operad matrices, it leads to the notion of local cocycle $n$-Lie bialgebras. Furthermore, we establish a one-to-one correspondence between the double of $n$-Lie bialgebras and Manin triples of $n$-Lie algebras.

math.RA↗
arXiv2022

How to construct Gorenstein projective modules relative to complete duality pairs over Morita rings

Let $Δ=\left(\begin{smallmatrix} A & {_AN_B}\\ {_BM_A} & B \\\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_{A}N=0=N\otimes_{B}M$.We first study how to construct (complete) duality pairs of $Δ$-modules using (complete) duality pairs of $A$-modules and $B$-modules, generalizing the result of Mao (Comm. Algebra, 2020, 12: 5296--5310) about the duality pairs over a triangular matrix ring. Moreover, we construct Gorenstein projective modules relative to complete duality pairs of $Δ$-modules. Finally, we give an application to Ding projective modules.

math.RA↗
arXiv2017

Enveloping algebras of double Poisson-Ore extensions

It is proved that the Poisson enveloping algebra of a double Poisson-Ore extension is an iterated double Ore extension. As an application, properties that are preserved under iterated double Ore extensions are invariants of the Poisson enveloping algebra of a double Poisson-Ore extension.

math.RA↗
arXiv2016

Homological unimodularity and Calabi-Yau condition for Poisson algebras

In this paper, we show that the twisted Poincaré duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives another definition of a unimodular Poisson algebra in terms of its Poisson Picard group. We also achieve twisted Poincaré duality for Hochschild (co)homology of Poisson bimodules using rigid dualizing complex. For a smooth Poisson affine variety with the trivial canonical bundle, we prove that its enveloping algebra is a Calabi-Yau algebra if the Poisson structure is unimodular.

math.RA↗
arXiv2014

Universal enveloping algebras of differential graded Poisson algebras

In this paper, we introduce the notion of differential graded Poisson algebra and study its universal enveloping algebra. From any differential graded Poisson algebra $A$, we construct two isomorphic differential graded algebras: $A^e$ and $A^E$. It is proved that the category of differential graded Poisson modules over $A$ is isomorphic to the category of differential graded modules over $A^e$, and $A^e$ is the unique universal enveloping algebra of $A$ up to isomorphisms. As applications of the universal property of $A^e$, we prove that $(A^e)^{op}\cong (A^{op})^e$ and $(A\otimes_{\Bbbk}B)^e\cong A^e\otimes_{\Bbbk}B^e$ as differential graded algebras. As consequences, we obtain that ``$e$'' is a monoidal functor and establish links among the universal enveloping algebras of differential graded Poisson algebras, differential graded Lie algebras and associative algebras.

math.RA↗
arXiv2014

Universal enveloping algebras of Poisson Ore extensions

We prove that the universal enveloping algebra of a Poisson-Ore extension is a length two iterated Ore extension of the original universal enveloping algebra. As consequences, we observe certain ring-theoretic invariants of the universal enveloping algebras that are preserved under iterated Poisson-Ore extensions. We apply our results to iterated quadratic Poisson algebras arising from semiclassical limits of quantized coordinate rings and a family of graded Poisson algebras of Poisson structures of rank at most two.

math.RA↗
arXiv2014

Universal enveloping algebras of Poisson Hopf algebras

For a Poisson algebra $A$, by exploring its relation with Lie-Rinehart algebras, we prove a Poincaré-Birkoff-Witt theorem for its universal enveloping algebra $A^e$. Some general properties of the universal enveloping algebras of Poisson Hopf algebras are studied. Given a Poisson Hopf algebra $B$, we give the necessary and sufficient conditions for a Poisson polynomial algebra $B[x; α, δ]_p$ to be a Poisson Hopf algebra. We also prove a structure theorem for $B^e$ when $B$ is a pointed Poisson Hopf algebra. Namely, $B^e$ is isomorphic to $B#_σ\mathcal{H}(B)$, the crossed product of $B$ and $\mathcal{H}(B)$, where $\mathcal{H}(B)$ is the quotient Hopf algebra $B^e/B^eB^+$.

math.RA↗