Related papers: A Hopf algebra associated to a Lie pair

We studied both the double cross product and the bicrossproduct constructions for the Hom-Hopf algebras of general $(\alpha,\beta)$-type. This allows us to consider the universal enveloping Hom-Hopf algebras of Hom-Lie algebras, which are…

We study quantization of a class of inhomogeneous Lie bialgebras which are crossproducts in dual sectors with Abelian invariant parts. This class forms a category stable under dualization and the double operations. The quantization turns…

We construct an universal enveloping algebra associated to the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincar\'e-Birkhoff-Witt theorem is proven is this context. It this then shown that this universal…

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase…

A weakly complete vector space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ is isomorphic to $\mathbb{K}^X$ for some set $X$ algebraically and topologically. The significance of this type of topological vector spaces is…

The correspondence between Lie algebras, Lie groups, and algebraic groups, on one side and commutative Hopf algebras on the other side are known for a long time by works of Hochschild-Mostow and others. We extend this correspondence by…

In this article character groups of Hopf algebras are studied from the perspective of infinite-dimensional Lie theory. For a graded and connected Hopf algebra we construct an infinite-dimensional Lie group structure on the character group…

We define wreath products of cocommutative Hopf algebras, and show that they enjoy a universal property of classifying cleft extensions, analogous to the Kaloujnine-Krasner theorem for groups. We show that the group ring of a wreath product…

A natural extension of the Hopf-cyclic cohomology, with coefficients, is introduced to encompass topological Hopf algebras. The topological theory allows to work with infinite dimensional Lie algebras. Furthermore, the category of…

Let $A \subseteq E$ be a given extension of Hopf (respectively Lie) algebras. We answer the \emph{classifying complements problem} (CCP) which consists of describing and classifying all complements of $A$ in $E$. If $H$ is a given…

Let $Q$ be a finite type quiver i.e. ADE Dynkin quiver. Denote by $\Lambda$ its preprojective algebra. It is known that there are finitely many indecomposable $\Lambda$-modules if and only if $Q$ is of type $A_1,A_2,A_3,A_4$. In this paper,…

Hopf algebroids are generalization of Hopf algebras over non-commutative base rings. It consists of a left- and a right-bialgebroid structure related by a map called the antipode. However, if the base ring of a Hopf algebroid is commutative…

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 $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$…

For a Poisson algebra $A$, by exploring its relation with Lie-Rinehart algebras, we prove a Poincar\'e-Birkoff-Witt theorem for its universal enveloping algebra $A^e$. Some general properties of the universal enveloping algebras of Poisson…

We study the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case. We prove that it admits a graded, finte-dimensional, connected gradation. Dually, the vector space IR is both a pre-Lie algebra for the…

In this work, the notion of partial representation of a Hopf algebra is introduced and its relationship with partial actions of Hopf algebras is explored. Given a Hopf algebra $H$, one can associate it to a Hopf algebroid $H_{par}$ which…

In this paper we generalize classical results on Lie algebras and universal enveloping algebras of Lie algebras to Lie-Rinehart algebras. We define for any Lie-Rinehart algebra $L$ and any cocycle $f$ in $Z^2(L,B)$, a universal enveloping…

We study universal enveloping Hopf algebras of Lie algebras in the category of weakly complete vector spaces over the real and complex field.

We prove that a Hopf algebra of prime dimension $p$ over an algebraically closed field, whose characteristic is equal to $p$, is either a group algebra or a restricted universal enveloping algebra. Moreover, we show that any Hopf algebra of…

Let ${U}_q(sl_2)$ be the quantized enveloping algebra associated to the simple Lie algebra $sl_2$. In this paper, we study the quantum double $D_q$ of the Borel subalgebra ${U}_q((sl_2)^{\leq 0})$ of ${U}_q(sl_2)$. We construct an analogue…