A spectral theory for transverse tensor operators

Abstract

We prove that tensor spaces over Lie algebras -rather than over associative rings -are universal amongst all linearly constrained tensor spaces. These Lie algebras compress the ambient tensor space of several well-known tensors, including matrix multiplication, quantum states in physics, relational data tensors, chat-room tensors, simple and Azumaya algebras, and simple Lie modules. This gives structural insights for tensors and improves how we recognize tensor when given in arbitrary bases.Our method builds a spectral theory of transverse operators developing a correspondence of Galois type between tensors, polynomials, and transverse operators. It also permits us to characterize the group actions on tensors using toric schemes. We prove polynomial-time algorithms for this correspondence and provide companion software packages for the computer algebra system Magma.

Tensor Isomorphism by conjugacy of Lie algebras

“…The rings used were each defined by corresponding universal properties. However, the recent work of [9] places the theory in a broader context, and demonstrates that universality in this setting involves Lie algebras (derivations) and their tensor space analogues (densors). This is the theory that underpins our new isomorphism test, which we call the derivation-densor method.…”

“…First, the derivation algebras L i := Der(t i ) (Line 1) are constructed in polynomial time using [9,Theorem F]. By Elton's Lemma, each L i is reductive which allows us to decompose Theorem 1] and the more general finite field case discussed in [15, pp.…”

“…Previous attacks on the problem also exploit ideas from a variety of areas, including * -algebras, Lie and Jordan algebras, numerical methods, and invariant theory. The method we propose here uses a recently discovered ternary Galois connection [9] to generalize several existing approaches to the problem.…”