Jiafeng Lü

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.

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.

DG Poisson algebra and its universal enveloping algebra

We introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let A be any DG Poisson algebra. We construct the universal enveloping algebra of A explicitly, which is denoted by Aue. We show that Aue has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over A is isomorphic to the category of DG modules over Aue. Furthermore, we prove that the notion of universal enveloping algebra Aue is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.

Quasi-Piecewise-Koszul Modules

The main aim of this article is to discuss the piecewise-Koszul property of finitely generated modules in the nongraded case. In particular, the notion of quasi-piecewise-Koszul module, a natural extension of quasi-Koszul modules (see [2]), quasi-d-Koszul modules (see [12] or [6]) and piecewise-Koszul modules (see [7]), is introduced. Let R be a Noetherian semiperfect algebra with Jacobson radical J and M a finitely generated R-module. The structure of the graded -module is studied in detail and some necessary and sufficient conditions for a finitely generated R-module to be quasi-piecewise-Koszul are provided. Moreover, as an application of quasi-piecewise-Koszul modules, we give a necessary and sufficient condition for the minimal Horseshoe Lemma to be true in the category of quasi-piecewise-Koszul modules, which perfects Theorem 2.8 of [13] and Theorem 3.1 of [9]. Finally, some applications of the minimal Horseshoe Lemma are also given.

Enveloping algebras of double Poisson-Ore extensions

ABSTRACT 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.

The structures on the universal enveloping algebras of differential graded Poisson Hopf algebras

ABSTRACT In this paper, the so-called differential graded (DG for short) Poisson Hopf algebra is introduced, which can be considered as a natural extension of Poisson Hopf algebras in the differential graded setting. The structures on the universal enveloping algebras of differential graded Poisson Hopf algebras are discussed.

Poincaré-Birkhoff-Witt Deformations of a Class of N-Homogeneous Algebras

In this paper, we mainly focus on the Poincaré–Birkhoff–Witt (PBW) deformation theory for a class of N-homogeneous algebras; here N ≥ 2 is an integer, which generalizes the results in [2] and [7]. More precisely, let k be a field of characteristic zero, V a finite dimensional vector space over k, and A = T(V)/(R) an N-homogeneous algebra (i.e., R ⊆ V ⊗N ) with being supported in a single degree d such that d > N. Set . Assume that P is a subspace of F N and (P) is a two sided ideal of T(V). Let U = T(V)/(P) be the deformation algebra of A. It is proved that U is the PBW-deformation of A if and only if J n ∩ F n−1 = J n−1 for any N ≤ n ≤ d. And if in particular d ≤ 2N, then U is the PBW-deformation of A if and only if P ∩ F N−1 = 0 and J d ∩ F d−1 = J d−1, where for n ≥ 0 and J n = 0 for n < N.

δ-Koszulity of Finitely Generated Graded Modules

The δ-Koszulity of finitely generated graded modules is discussed and the notion of weakly δ-Koszul module is introduced. Let M ∈ gr(A) and {S d 1 , S d 2 ,…, S d m } denote the set of minimal homogeneous generating spaces of M where S d i consists of homogeneous elements of M of degree d i . Put ℳ1 = ⟨ S d 1 ⟩, ℳ2 = ⟨ S d 1 , S d 2 ⟩,…, ℳ m = ⟨ S d 1 , S d 2 ,…, S d m ⟩. Then M admits a chain of graded submodules: 0 = ℳ0 ⊂ ℳ1 ⊂ ℳ2 ⊂ … ⊂ ℳ m = M. Moreover, it is proved that M is a weakly δ-Koszul module if and only if all ℳ i /ℳ i−1[−d i ] are δ-Koszul modules, if and only if the associated graded module G(M) is a δ-Koszul module. Further, as applications, the relationships of minimal graded projective resolutions among M, G(M) and these quotients ℳ i /ℳ i−1 are established. The Ext module of a weakly δ-Koszul module M is proved to be finitely generated in degree zero.