Gunnar Fløystad, Dominique Manchon, Hans Z. Munthe-Kaas
We organize colored aromatic trees into a pre-Lie-Rinehart algebra (i.e. a flat torsion-free Lie-Rinehart algebra) endowed with a natural trace map, and show the freeness of this object among pre-Lie-Rinehart algebras with trace. This yields the algebraic foundations of aromatic B-series.
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The free tracial post-Lie-Rinehart algebra of planar aromatic trees for the design of divergence-free Lie-group methods
本文基于平面芳香树研究无散度积分器，推导出保持几何无散度特性的高阶李群方法。
In this paper we study the universal central extension of a Lie--Rinehart algebra and we give a description of it. Then we study the lifting of automorphisms and derivations to central extensions. We also give a definition of a non-abelian tensor product in Lie--Rinehart algebras based on the construction of Ellis of non-abelian tensor product of Lie algebras. We relate this non-abelian tensor product to the universal central extension.
We prove how the universal enveloping algebra constructions for Lie-Rinehart algebras and anchored Lie algebras are naturally left adjoint functors. This provides a conceptual motivation for the universal properties these constructions satisfy. As a supplement, the categorical approach offers new insights into the definitions of Lie-Rinehart algebra morphisms, of modules over Lie-Rinehart algebras and of the infinitesimal gauge algebra of a module.
In this paper we study universal central extensions and non-abelian tensor product of hom-Lie-Rinehart algebras. We discuss about universal $α$- central extensions, and, lifting of automorphisms and $α$-derivations to central extensions for hom-Lie-Rinehart algebras. This is in turn provide such lifting of automorphisms and $α^k$-derivations to the central extensions for hom-Lie algebras.
Lie-Rinehart algebras, also known as Lie algebroids, give rise to Hopf algebroids by a universal enveloping algebra construction, much as the universal enveloping algebra of an ordinary Lie algebra gives a Hopf algebra, of infinite dimension. In finite characteristic, the universal enveloping algebra of a restricted Lie algebra admits a quotient Hopf algebra which is finite-dimensional if the Lie algebra is. Rumynin has shown that suitably defined restricted Lie algebroids allow to define restricted universal enveloping algebras that are finitely generated projective if the Lie algebroid is. T…
Post-Hopf algebroids, post-Lie-Rinehart algebras and geometric numerical integration
本文提出后霍普夫余代数概念，构造作用后霍普夫余代数，并展示后李-里纳尔特代数的 universal enveloping 代数性质，应用于流形上的几何数值积分。
Aromatic and clumped multi-indices: algebraic structure and Hopf embeddings
本文引入芳香和聚类多指标，通过代数结构简化体积保持方法的研究，同时保留原有结构，扩展霍普夫嵌入至芳香情境。
Universal enveloping algebras of Lie-Rinehart algebras: crossed products, connections, and curvature
本文将 Blattner-Cohen-Montgomery 定理推广至左 Hopf 代数胚，给出射影 Lie-Rinehart 代数泛包络代数的交叉积分解，并应用于主丛上的几何例子。
We consider Lie algebroids over algebraic spaces (in short we call it as $a$-spaces) by considering the sheaf of Lie-Rinehart algebras. We discuss about properties of universal enveloping algebroid $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$ of a Lie algebroid $\mathcal{L}$ over an $a$-space $(X, \mathcal{O}_X)$. This is done by sheafification of the presheaf of universal enveloping algebras for Lie-Rinehart algebras. We review the extent to which structure of the universal enveloping algebroid of Lie algebroids (over special $a$-spaces) resembles a sheaf of bialgebras. In the sequel we present a…
Hochschild cohomology of Lie-Rinehart algebras
本文通过非线性 Chevalley–Eilenberg 上链复形及推广的 PBW 同构，用相关分次商代数的 Poisson 上同调计算 Lie-Rinehart 代数泛包络代数的 Hochschild 上同调，并推导对偶 Hochschild 与循环同调结果。
In this paper, we define and study the universal enveloping algebra of a Poisson superalgebra. In particular, a new PBW Theorem for Lie-Rinehart superalgebras is proved leading to a PBW Theorem for Poisson superalgebras, we show the universal enveloping algebra of a Poisson Hopf superalgebra (resp. Poisson-Ore extension) is a Hopf superalgebra (resp. iterated Ore extension), and we study the universal enveloping algebra for interesting classes of Poisson superalgebras such as Poisson symplectic superalgebras.
On Poisson superalgebras in characteristic 2
本文研究特征为2的泊松超代数，探讨其表示、上同调、变形等，并引入了特征为2的李-里纳尔特超代数，揭示其与泊松超代数的联系。
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 $\m…
In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this notion to define the universal enveloping algebra of the category of $\tilde{L}$-connections and to define the cohomology and homology of an arbitrary connection. In this note we introduce the canonical quotient $L$ of a D-Lie algebra $\tilde{L}$ and use this to classify D-Lie algebras where $L$ is projective as $A$-module. We define for any 2-co…
Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the s…
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