Tensor Categories: Unifying Language of Symmetry

Why This Matters

Explores connections: TC ↔ Rep/Alg-Comb, AG/NT, Topology, Operator Algebras, VOAs.

Warm-up Categories

• **Vec_k**: tensor ⊗, unit 1, duals, symmetry.
• **Vec_G**: simples k_g, fusion rules, dual g⁻¹.
• **Rep(G)**: rigid, symmetric; Maschke (char 0).
• Edge to Semisimple Fusion: Fusion rules generalize character arithmetic.

Tannakian Snapshot

Tannaka (Deligne–Milne): fiber functor ω: C→Vec; G=Aut^⊗(ω); C≅Rep(G).

Symmetric Tensor Categories

Symmetric tensor categories with fiber functor ↔ affine group schemes.

Influences

• Saavedra-Rivano ’72
• Deligne (neutral Tannakian)

Number Theory/AG Edge

Fundamental group schemes, motives connect to number theory and algebraic geometry.

Applications of Tensor Categories

Impact on representation theory, quantum physics, and mathematical structures.

Future Directions

Exploring new connections and applications in mathematics and physics.