Toward Jordan Decompositions of Tensors

TL;DR

This paper develops a framework linking tensor invariants to Jordan decompositions via Lie algebra actions, enabling improved tensor classification and insights into quantum information.

Contribution

It introduces a novel algebraic approach to associate Jordan decompositions with tensors through Lie algebra embeddings, enhancing tensor analysis methods.

Findings

Establishes a consistent group action on tensors and their adjoint operators.

Shows invariants of the adjoint operator are tensor invariants.

Provides applications to tensor decomposition and quantum information classification.

Abstract

We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra that acts on it and embed their direct sum into an auxiliary algebra. Viewed as endomorphisms of this algebra, we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which means that the invariants of the adjoint operator of a tensor, such as the Jordan decomposition, are invariants of the tensor. We show that there is an essentially unique algebra structure that preserves the tensor structure and has a meaningful Jordan decomposition. We utilize aspects of these adjoint operators to study orbit separation and classification in examples relevant to tensor decomposition and quantum information.