Invariant theory and 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.