Tensor Triangular Geometry for Quantum Groups

Let $\mathfrak g$ be a complex simple Lie algebra and let $U_{\zeta}({\mathfrak g})$ be the corresponding Lusztig ${\mathbb Z}[q,q^{-1}]$-form of the quantized enveloping algebra specialized to an $\ell$th root of unity. Moreover, let $\mod(U_{\zeta}({\mathfrak g}))$ be the braided monoidal category of finite-dimensional modules for $U_{\zeta}({\mathfrak g})$. In this paper we classify the thick tensor ideals of $\mod(U_{\zeta}({\mathfrak g}))$ and compute the prime spectrum of the stable module category associated to $\text{mod}(U_{\zeta}({\mathfrak g}))$ as defined by Balmer.