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 V=T(V)⊗T(V⋆), where T(V) is the tensor algebra on an n-dimensional vector space over a field K of characteristic 0. First we classify all the ideals of the initial object Z in the category of wheeled PROPs. We show that non-degenerate sub-wheeled PROPs of V are exactly subalgebras of the form VG where G is a closed, reductive subgroup of the general linear group 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×n matrix ring over a commutative algebra R. Namely, we prove that a wheeled PROP can be embedded in R⊗V for a commutative K-algebra R if and only if it satisfies certain relations.