Exact sequences of inner automorphisms of tensors

Abstract

We produce a long exact sequence whose terms are unit groups of associative algebras that behave as inner automorphisms of a given tensor. Our sequence generalizes known sequences for associative and non-associative algebras. In a manner similar to those, our sequence facilitates inductive reasoning about, and calculation of the groups of symmetries of a tensor. The new insights these methods afford can be applied to problems ranging from understanding algebraic structures to distinguishing entangled states in particle physics.In memory of C.C. Sims.

Tensor Isomorphism by conjugacy of Lie algebras

“…In general, one expects that Der(t) will contain many proper ideals. For example, each kernel of the (exponential number of) homomorphisms described in the exact sequences of [1] yield proper ideals of Der(t). However, modular Lie theory can become quite involved.…”

“…Let K be a field with either K = 6K finite or K/Q finite. There is an algorithm that, given s, t ∈ (K d‫ו‬ ⊗ • • • ⊗ K d1 ) * where Der(s) has Chevalley type and dim s = 1, decides if s ∼ = t using (d ‫ו‬ + • • • + d 1 ) O (1) steps.…”