Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators

Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrödinger algebras

We discuss a procedure to determine finite sets $\mathcal{M}$ within the commutant of an algebraic Hamiltonian in the enveloping algebra of a Lie algebra $\mathfrak{g}$ such that their generators define a quadratic algebra. Although independent from any realization of Lie algebras by differential operators, the method is partially based on an analytical approach, and uses the coadjoint representation of the Lie algebra $\mathfrak{g}$. The procedure, valid for non-semisimple algebras, is tested for the centrally extended Schrödinger algebras $\widehat{S}(n)$ for various different choices of alg…

Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras

We propose a systematic procedure to construct polynomial algebras from intermediate Casimir invariants arising from (semisimple or non-semisimple) Lie algebras $\mathfrak{g}$. In this approach, we deal with explicit polynomials in the enveloping algebra of $\mathfrak{g} \oplus \mathfrak{g} \oplus \mathfrak{g}$. We present explicit examples to show how these Lie algebras can display different behaviours and can lead to Abelian algebras, quadratic algebras or more complex structures involving higher order nested commutators. Within this framework, we also demonstrate how virtual copies of the L…

Plancherel Theorem and the Left Ideals of the Group Algebra for the Jacobi Group

Let G be the three dimensional connected real semisimple Lie group and let KAN be the Iwasawa decomposition of G.Let J be the Jacobi group, which is the semidirect product of the two groups Heisenberg group with G. The Jacobi group plays an important role in Quantum Mechanics. The purpose of this paper is to define the Fourier transform in order to obtain the Plancherel theorem for the group J. To this end a classification of all left ideals of the group algebra for the Jacobi group

$\mathfrak{sl}(2,\mathbb{C})$ 的通用包络代数的有限维李子代数

Finite-dimensional Lie subalgebras of the universal enveloping algebra of $\mathfrak{sl}(2,\mathbb{C})$

本文完整分类了特殊线性李代数sl(2,C)通用包络代数中的所有非零有限维复数李子代数，证明幂零子代数必为阿贝尔、不可解子代数为约化且半单部分同构于sl(2,C)，非幂零可解子代数为阿贝尔中心与可解族r(d,m)的直和，并讨论了实斜埃尔米特情形及其与外尔代数实现的关系。

The algebraic structure of Galilean superconformal symmetries

The semisimple part of d-dimensional Galilean conformal algebra g^(d) is given by h^(d)=O(2,1)+O(d), which after adding via semidirect sum the 3d-dimensional Abelian algebra t^(d) of translations, Galilean boosts and constant accelerations completes the construction. We obtain Galilean superconformal algebra G^(d) by firstly defining the semisimple superalgebra H^(d) which supersymmetrizes h^(d), and further by considering the expansion of H^(d) by tensorial and spinorial graded Abelian charges in order to supersymmetrize the Abelian generators of t^(d). For d=3 the supersymmetrization of h^(3…

Central Extensions of the Quasi-orthogonal Lie algebras

We determine the central extensions of a whole family of Lie algebras, obtained by the method of graded contractions from so(N+1), N arbitrary. All the inhomogeneous orthogonal and pseudo-orthogonal algebras are members of this family, as well as a large number of other non-semisimple algebras, all of which have at least a semidirect structure (in some cases two or more). The dimensions of their second cohomology groups H^2(G,R) and the explicit expression of their central extensions are given.