On structure of homogenenous Wick ideals in Wick $*$-algebras with braided coefficients

TL;DR

This paper investigates the structure of homogeneous Wick ideals in quadratic Wick algebras with braided coefficients, providing a recursive construction method and analyzing their representations.

Contribution

It introduces a method to construct higher-degree homogeneous Wick ideals from lower-degree ones and describes the largest such ideals in specific cases.

Findings

Recursive construction of Wick ideals of higher degrees

Description of largest homogeneous Wick ideals in certain cases

Analysis of *-representations annihilating specific ideals

Abstract

We study the structure of Wick homogenenous ideals of higher degrees in quadratic algebras allowing Wick ordering. We present a method how to construct a homogeneous Wick ideal $\mathcal{I}_{n+1}$ of degree $n+1$ out of a homogeneous Wick ideal $\mathcal{I}_n$ of degree $n$ so that $\mathcal{I}_{n+1}\subset\mathcal{I}_n$. We show that in some particular cases our procedure allows one to get a description of the largest homogeneous Wick ideals of higher degrees having generators of the largest quadratic Wick ideal only. Finally we study classes of $*$-representations of Wick version of CCR annihilating certain homogeneous Wick ideals of degree higher than $2$.