Deformations of the Boson $sp(4,R)$ Representation and its Subalgebras

TL;DR

This paper explores deformations of the boson representation of the sp(4,R) algebra, including q-deformations, and analyzes their subalgebras, revealing new reducible and irreducible structures within the same Fock space.

Contribution

It introduces two distinct deformations of the sp(4,R) boson representation and examines their subalgebras, providing a detailed algebraic and representation-theoretic analysis.

Findings

Deformed representations act in the same Fock space.

Subalgebras decompose into irreducible representations.

New tensor-based deformation relates to angular momentum algebra.

Abstract

The boson representation of the sp(4,R) algebra and two distinct deformations of it, are considered, as well as the compact and noncompact subalgebras of each. The initial as well as the deformed representations act in the same Fock space. One of the deformed representation is based on the standard q-deformation of the boson creation and annihilation operators. The subalgebras of sp(4,R) (compact u(2) and three representations of the noncompact u(1,1) are also deformed and are contained in this deformed algebra. They are reducible in the action spaces of sp(4,R) and decompose into irreducible representations. The other deformed representation, is realized by means of a transformation of the q-deformed bosons into q-tensors (spinor-like) with respect to the standard deformed su(2). All of its generators are deformed and have expressions in terms of tensor products of spinor-like operators. In this case, an other deformation of su(2) appears in a natural way as a subalgebra and can be interpreted as a deformation of the angular momentum algebra so(3). Its representation is reducible and decomposes into irreducible ones that yields a complete description of the same.