Invariant theory and wheeled PROPs

TL;DR

This paper explores the structure of wheeled PROPs using Invariant Theory, classifying ideals, describing invariant subalgebras, and generalizing Procesi's theorem to embed wheeled PROPs into matrix rings over commutative algebras.

Contribution

It provides a classification of ideals in the initial wheeled PROP and characterizes invariant sub-wheeled PROPs as subalgebras fixed by reductive groups, extending classical invariant results.

Findings

Classified all ideals of the initial wheeled PROP.

Described invariant sub-wheeled PROPs as fixed subalgebras under reductive groups.

Extended Procesi's embedding theorem to wheeled PROPs.

Abstract

We study the category of wheeled PROPs using tools from Invariant Theory. A typical example of a wheeled PROP is the mixed tensor algebra ${\mathcal V}=T(V)\otimes T(V^\star)$, where $T(V)$ is the tensor algebra on an $n$-dimensional vector space over a field of $K$ of characteristic 0. First we classify all the ideals of the initial object ${\mathcal{Z}}$ in the category of wheeled PROPs. We show that non-degenerate sub-wheeled PROPs of ${\mathcal V}$ are exactly subalgebras of the form ${\mathcal V}^G$ where $G$ is a closed, reductive subgroup of the general linear group ${\rm GL}(V)$. When $V$ is a finite dimensional Hilbert space, a similar description of invariant tensors for an action of a compact group was given by Schrijver. We also generalize the theorem of Procesi that says that trace rings satisfying the $n$-th Cayley-Hamilton identity can be embedded in an $n \times n$ matrix ring over a commutative algebra $R$. Namely, we prove that a wheeled PROP can be embedded in $R\otimes {\mathcal V}$ for a commutative $K$-algebra $R$ if and only if it satisfies certain relations.