Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry

Understand

We introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group.

• We show that an algebraic higher symmetry in a bosonic system in $n$-dimensional space is characterized and classified by a local fusion $n$-category.
• We find another way to describe algebraic higher symmetry by restricting to symmetric sub Hilbert space where symmetry transformations all become trivial.
• In this case, algebraic higher symmetry can be fully characterized by a non-invertible gravitational anomaly (i.e.