Quantum affine algebras at roots of unity

Abstract:

Let $U_{q}(\hat {\mathfrak {g}})$ be the quantized universal enveloping algebra of the affine Lie algebra $\hat {\mathfrak {g}}$ associated to a finite-dimensional complex simple Lie algebra $\mathfrak {g}$, and let $U_{q}^{\mathrm {res}}(\hat {\mathfrak {g}})$ be the $\mathbb {C}[q,q^{-1}]$-subalgebra of $U_{q}(\hat {\mathfrak {g}})$ generated by the $q$-divided powers of the Chevalley generators. Let $U_{\epsilon }^{\mathrm {res}}(\hat {\mathfrak {g}})$ be the Hopf algebra obtained from $U_{q}^{\mathrm {res}}(\hat {\mathfrak {g}})$ by specialising $q$ to a non-zero complex number $\epsilon$ of odd order. We classify the finite-dimensional irreducible representations of $U_{\epsilon }^{\mathrm {res}}(\hat {\mathfrak {g}})$ in terms of highest weights. We also give a “factorisation” theorem for such representations: namely, any finite-dimensional irreducible representation of $U_{\epsilon }^{\mathrm {res}}(\hat {\mathfrak {g}})$ is isomorphic to a tensor product of two representations, one factor being the pull-back of a representation of $\hat {\mathfrak {g}}$ by Lusztig’s Frobenius homomorphism $\hat {\mathrm {Fr}}_{\epsilon }:U_{\epsilon }^{\mathrm {res}}(\hat {\mathfrak {g}})\to U(\hat {\mathfrak {g}})$, the other factor being an irreducible representation of the Frobenius kernel. Finally, we give a concrete construction of all of the finite-dimensional irreducible representations of $U_{\epsilon }^{\mathrm {res}}(\hat {sl}_{2})$. The proofs make use of several interesting new identities in $U_{q}(\hat {\mathfrak {g}})$.