Related papers: Quadratic Lie Algebras

Using the concept of mixable shuffles, we formulate explicitly the quantum quasi-shuffle product. We also provide a desirable description of the subalgebra generated by the set of primitive elements of the quantum quasi-shuffle bialgebra. A…

General braided counterparts of classical Clifford algebras are introduced and investigated. Braided Clifford algebras are defined as Chevalley-Kahler deformations of the corresponding braided exterior algebras. Analogs of the spinor…

This paper is devoted to understanding the defining ideal of a Nichols algebra from the decomposition of specific elements in the group algebra of braid groups. A family of primitive elements are found and algorithms are proposed. To prove…

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…

Let A be a basic finite dimensional and connected algebra over an algebraically closed field k with zero characteristic. If the ordinary quiver of A has no double bypasses, we show that A admits a Galois covering which satisfies a universal…

The construction of superintegrable systems based on Lie algebras and their universal enveloping algebras has been widely studied over the past decades. However, most constructions rely on explicit differential operator realisations and…

The purpose of this paper is to make the theory of vertex algebras trivial. We do this by setting up some categorical machinery so that vertex algebras are just ``singular commutative rings'' in a certain category. This makes it easy to…

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…

We provide spectral Lie algebras with enveloping algebras over the operad of little $G$-framed $n$-dimensional disks for any choice of dimension $n$ and structure group $G$, and we describe these objects in two complementary ways. The first…

In this paper we give a construction of the universal enveloping algebra of a Malcev algebra in categories of group algebra comodules with a symmetry given by a bicharacter of the group. A particular example of such categories is the…

In this dissertation, we investigate the cohomology theory of restricted Lie algebras. The representation theory of restricted Lie algebras is reviewed including a description of the restricted universal enveloping algebra. In the case of…

We give a geometric realization of the whole universal enveloping algebra of the Borcherds-Bozec algebra using quiver with loops via the motivic semi-derived Hall algebra approach. In particular, using acyclic quivers, we give a geometric…

In this paper we define two Lie operations, and with that we define the bicharacter algebras, Nichols bicharacter algebras, quantum Nichols bicharacter algebras, etc. We obtain explicit bases for $\mathfrak L(V)${\tiny $_{R}$} and…

In this paper we present an approach to quadratic structures in derived algebraic geometry. We define derived n-shifted quadratic complexes, over derived affine stacks and over general derived stacks, and give several examples of those. We…

We investigate the Nichols algebra $\mathfrak{B}(V_{abe})$ which are from the Yetter-Drinfeld category of Suzuki algebras. The $4n$ and $n^2$ dimensional Nichols algebras, first appeared in \cite{Andruskiewitsch2018}, are obtained again via…

It is shown that if the universal enveloping algebra of a simple $\mathbb Z^n$-graded Lie algebra is Noetherian, then the Lie algebra is finite-dimensional.

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…

Let $\mathfrak{g}$ be a finite dimensional complex simple classical Lie superalgebra and $A$ be a commutative, associative algebra with unity over $\mathbb{C}$. In this paper we define an integral form for the universal enveloping algebra…

A Lie superalgebra endowed with a supersymmetric, even, non-degenerate, invariant bilinear form is called a quadratic Lie superalgebra. In this paper we give inductive descriptions of quadratic Lie superalgebras in terms of generalized…

This work is prompted by the long standing question of whether it is possible for the universal enveloping algebra of an infinite dimensional Lie algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn…