The Lie algebra of the lowest transitively differential group of degree three

Abstract

We investigate the real Lie algebra of first-order differential operators with polynomial coefficients, which is subject to the following requirements. (1) The Lie algebra should admit a basis of differential operators with homogeneous polynomial coefficients of degree up to and including three. (2) The generator of the algebra must include the translation operators ${\partial }_k$ for all the variables $x_1$,...,$x_k$. (3) The Lie algebra is the smallest indecomposable Lie algebra satisfying (1) and (2). It turns out to be a 39-dimensional Lie algebra in six variables ($k=6$) and the construction of this algebra is also the simplest possible case in the general construction of the Lie algebras of the transitively differential groups introduced by Guillemin and Sternberg in 1964 involving the coefficients of degree 3. Those algebras and various subalgebras have similarities with algebras related to different applications in physics such as those of the Schrödinger, Conformal and Galilei transformation groups with and without central extension. The paper is devoted to the presentation of the structure and different decompositions of the Lie algebra under investigation. It is also devoted to the presentation of relevant Lie subalgebras and the construction of their Casimir invariants using different methods. We will rely, in particular, on differential operator realizations, symbolic computation packages, the Berezin bracket and virtual copies of the Lie algebras.

References in corpus (7)

• A class of solvable Lie algebras and their Casimir Invariants
• Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators
• Commutativity of missing label operators in terms of Berezin brackets
• Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrödinger algebras
• Lowest Weight Representations of Super Schrodinger Algebras in One Dimensional Space
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• Decomposition of enveloping algebras of simple Lie algebras and their related polynomial algebras