Centrally Extended Conformal Galilei Algebras and Invariant Nonlinear PDEs

We construct, for any given $\ell = \frac{1}{2} + {\mathbb N}_0,$ the second-order \textit{nonlinear} partial differential equations (PDEs) which are invariant under the transformations generated by the centrally extended conformal Galilei algebras. The generators are obtained by a coset construction and the PDEs are constructed by standard Lie symmetry technique. It is observed that the invariant PDEs have significant difference for $\ell > \frac{3}{2}.$Comment: 22pages, 3figure

On irreducible representations of the exotic conformal Galilei algebra

We investigate the representations of the exotic conformal Galilei algebra. This is done by explicitly constructing all singular vectors within the Verma modules, and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite dimensional irreducible modules is presented.Comment: 11 pages, added 6 references and conclusing remark

Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras

The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters $d$ and $\ell$. The aim of the present work is to investigate the lowest weight representations of CGA with $d = 1$ for any integer value of $\ell$. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if $\ell = 1$ and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when $\ell \neq 1$. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules

Highest weight representations and Kac determinants for a class of conformal Galilei algebras with central extension

We investigate the representations of a class of conformal Galilei algebras in one spatial dimension with central extension. This is done by explicitly constructing all singular vectors within the Verma modules, proving their completeness and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite dimensional irreducible modules is presented. It is also shown that a formula for the Kac determinant is deduced from our construction of singular vectors. Thus we prove a conjecture of Dobrev, Doebner and Mrugalla for the case of the Schrodinger algebra.Comment: 24 page

Irreducible representations of Z22\mathbb{Z}_2^2-graded supersymmetry algebra and their applications

We give a brief review on recent developments of $\mathbb{Z}_2^n$-graded symmetry in physics in which hidden $\mathbb{Z}_2^n$-graded symmetries and $\mathbb{Z}_2^n$-graded extensions of known systems are discussed. This elucidates physical relevance of the $\mathbb{Z}_2^n$-graded algebras. As an example of physically interesting algebra, we take $\mathbb{Z}_2^2$-graded supersymmetry (SUSY) algebras and consider their irreducible representations (irreps). A list of irreps for ${\cal N} = 1, 2$ algebras is presented and as an application of the irreps, $\mathbb{Z}_2^2$-graded SUSY classical actions are constructed

Hom-Lie-Virasoro symmetries in Bloch electron systems and quantum plane in tight binding models

We discuss the Curtright-Zachos (CZ) deformation of the Virasoro algebra and its extensions in terms of magnetic translation (MT) group in a discrete Bloch electron system, so-called the tight binding model (TBM), as well as in its continuous system. We verify that the CZ generators are essentially composed of a specific combination of MT operators representing deformed and undeformed U(1) translational groups, which determine phase factors for a ⁎-bracket commutator. The phase factors can be formulated as a ⁎-ordered product of the commutable U(1) operators by interpreting the AB phase factor of discrete MT action as fluctuation parameter q of a quantum plane. We also show that some sequences of TBM Hamiltonians are described by the CZ generators