A spectral theory for transverse tensor operators

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 {\sf Magma}.