Polynomial super-gl(n) algebras

TL;DR

This paper introduces a new class of finite-dimensional nonlinear superalgebras called polynomial super-$gl(n)$ algebras, exploring their structure, representations, and potential applications in quantum field theory.

Contribution

It defines and analyzes polynomial super-$gl(n)$ algebras with odd generators closing on polynomial functions of $gl(n)$, including explicit constructions and conditions for their consistency.

Findings

Defined a three-parameter family of quadratic super-$gl(n)$ algebras

Constructed Kac modules and discussed atypicality conditions

Explored applications in quantum field theory and supersymmetry

Abstract

We introduce a class of finite dimensional nonlinear superalgebras $L = L_{\bar{0}} + L_{\bar{1}}$ providing gradings of $L_{\bar{0}} = gl(n) \simeq sl(n) + gl(1)$. Odd generators close by anticommutation on polynomials (of degree $>1$) in the $gl(n)$ generators. Specifically, we investigate `type I' super-$gl(n)$ algebras, having odd generators transforming in a single irreducible representation of $gl(n)$ together with its contragredient. Admissible structure constants are discussed in terms of available $gl(n)$ couplings, and various special cases and candidate superalgebras are identified and exemplified via concrete oscillator constructions. For the case of the $n$-dimensional defining representation, with odd generators $Q_{a}, \bar{Q}{}^{b}$, and even generators ${E^{a}}_{b}$, $a,b = 1,...,n$, a three parameter family of quadratic super-$gl(n)$ algebras (deformations of $sl(n/1)$) is defined. In general, additional covariant Serre-type conditions are imposed, in order that the Jacobi identities be fulfilled. For these quadratic super-$gl(n)$ algebras, the construction of Kac modules, and conditions for atypicality, are briefly considered. Applications in quantum field theory, including Hamiltonian lattice QCD and space-time supersymmetry, are discussed.