Quantum Groups & Hopf Algebras

q-Parameter

q (real part): 0.90 q (imaginary phase): 0.00

At q=1: recover classical Lie algebra sl₂.
At q = root of unity: special finite-dimensional quotients appear — related to conformal field theory.

q-Numbers [n]_q

n	[n]_q	[n]!

[n]_q = (qⁿ - q⁻ⁿ)/(q - q⁻¹)
q-deformed integers. At q→1: [n]_q → n.

Hopf Algebra Structure

Uq(sl₂) generators: E, F, K, K⁻¹
Algebra:
KE = q² EK
KF = q⁻² FK
[E,F] = (K-K⁻¹)/(q-q⁻¹)
Coproduct Δ:
Δ(E) = E⊗K + 1⊗E
Δ(F) = F⊗1 + K⁻¹⊗F
Δ(K) = K⊗K
Antipode S:
S(E) = -EK⁻¹
S(F) = -KF
S(K) = K⁻¹

q-Deformed Representations

Spin j (half-integer): 1/2

The spin-j representation has dimension 2j+1. The q-deformed matrix elements use q-numbers.
ρ(E)|m⟩ = √([j-m]_q[j+m+1]_q) |m+1⟩

R-Matrix & Yang-Baxter

The quantum group provides a universal R-matrix satisfying the Yang-Baxter equation:
R₁₂ R₁₃ R₂₃ = R₂₃ R₁₃ R₁₂
This gives representations of the braid group — linking quantum groups to knot invariants (Jones polynomial).