A characterization of the unitary highest weight modules by Euclidean Jordan algebras

TL;DR

This paper characterizes unitary highest weight modules of the conformal algebra of Euclidean Jordan algebras, identifying a quadratic relation that characterizes modules with minimal Gelfand-Kirillov dimension and relates to the Joseph ideal.

Contribution

It introduces a specific quadratic relation in the universal enveloping algebra that characterizes certain unitary highest weight modules and connects this to the Joseph ideal.

Findings

Identifies a quadratic relation characterizing modules with minimal Gelfand-Kirillov dimension.

Finds a quadratic element in the universal enveloping algebra of the conformal algebra.

Shows the Joseph ideal contains this quadratic element.

Abstract

Let $\mathfrak{co}(J)$ be the conformal algebra of a simple Euclidean Jordan algebra $J$. We show that a (non-trivial) unitary highest weight $\mathfrak{co}(J)$-module has the smallest positive Gelfand-Kirillov dimension if and only if a certain quadratic relation is satisfied in the universal enveloping algebra $U(\mathfrak{co}(J)_{\mathbb{C}})$. In particular, we find an quadratic element in $U(\mathfrak{co}(J)_{\mathbb{C}})$. A prime ideal in $U(\mathfrak{co}(J)_{\mathbb{C}})$ equals the Joseph ideal if and only if it contains this quadratic element.