Related papers: A Casimir element inexpressible as a Lie polynomia…

We study the von Neumann algebra generated by q--deformed Gaussian elements l_i+l_i^* where operators l_i fulfill the q--deformed canonical commutation relations l_i l_j^*-q l_j^* l_i=delta_{ij} for -1<q<1. We show that if the number of…

We present a nonlinear realization of the 5-graded Lie algebra associated to a Kantor triple system. Any simple Lie algebra can be realized in this way, starting from an arbitrary 5-grading. In particular, we get a unified realization of…

Let $N$ be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra $\mathfrak{n}$ of dimension $n.$ Let $H$ be a subgroup of the automorphism group of $N.$ Assume that $H$ is a commutative, simply connected,…

A Lie polynomial is an element of a free Lie algebra $\mathcal F_k$ on $k$-generators, which defines a Lie map on a given Lie algebra $L$, by substituting $k$-elements of $L$. Similar to word maps on groups and polynomial maps on algebras,…

We study from an algebraic and geometric viewpoint Hamiltonian operators which are sum of a non-degenerate first-order homogeneous operator and a Poisson tensor. In flat coordinates, also known as Darboux coordinates, these operators are…

In this work, we consider Lie algebras L containing a subalgebra isomorphic to sl3 and such that L decomposes as a module for that sl3 subalgebra into copies of the adjoint module, the natural 3-dimensional module and its dual, and the…

An equivalent condition for an element of a Lie algebra acting nilpotently in all its representations is obtained. Namely, it should belong to the derived algebra and go via factoring over the radical to a nilpotent element of the…

In this paper we study non-nilpotent non-Lie Leibniz $\mathbb{F}$-algebras with one-dimensional derived subalgebra, where $\mathbb{F}$ is a field with $\operatorname{char}(\mathbb{F}) \neq 2$. We prove that such an algebra is isomorphic to…

We give a definition of quarternion Lie algebra and of the quarternification of a complex Lie algebra. By our definition gl(n,H), sl(n,H), so*(2n) and sp(n) are quarternifications of gl(n,C), sl(n,C), so(n,C) and u(n) respectively. Then we…

We consider the Casimir Invariants related to some a special kind of Lie-algebra extensions, called universal extensions. We show that these invariants can be studied using the equivalence between the universal extensions and the…

Construction of superintegrable systems based on Lie algebras have been introduced over the years. However, these approaches depend on explicit realisations, for instance as a differential operators, of the underlying Lie algebra. This is…

Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebra and root vectors and which make it possible to construct representations by operators acting…

Let $(H_{\mathbf{R}}, U_t)$ be any strongly continuous orthogonal representation of $\mathbf{R}$ on a real (separable) Hilbert space $H_{\mathbf{R}}$. For any $q\in (-1,1)$, we denote by $\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ the…

We introduce three non-trivial 2-cocycles $c_k$, k=0,1,2, on the Lie algebra $S^3H=Map(S^3,H)$ with the aid of the corresponding basis vector fields on $S^3$, and extend them to 2-cocycles on the Lie algebra $S^3gl(n,H)=S^3H \otimes…

As a homomorphic image of the hyperalgebra $U_{q,R}(m|n)$ associated with the quantum linear supergroup $U_\upsilon(\mathfrak{gl}_{m|n})$, we first give a presentation for the $q$-Schur superalgebra $S_{q,R}(m|n,r)$ over a commutative ring…

We study a class of representations of the Lie algebra of Laurent polynomials with values in the nilpotent subalgebra of sl(3). We derive Weyl-type (bosonic) character formulas for these representations. We establish a connection between…

Let $U_\varepsilon({\mathfrak g})$ be the standard simply connected version of the Drinfeld-Jumbo quantum group at an odd primitive m-th root of unity $\varepsilon$. The center of $U_\varepsilon({\mathfrak g})$ contains a huge commutative…

3-Lie algebras are constructed by Lie algebras, derivations and linear functions, associative commutative algebras, whose involutions and derivations. Then the 3-Lie algebras are obtained from group algebras $F[G]$. An infinite dimensional…

We present a unified ring theoretic approach, based on properties of the Casimir element of a symmetric algebra, to a variety of known divisibility results for the degrees of irreducible representations of semisimple Hopf algebras in…

The purpose of this article is to study the relationship between numerical invariants of certain subspace arrangements coming from reflection groups and numerical invariants arising in the representation theory of Cherednik algebras. For…