Representations of shifted twisted quantum affine algebras

Abstract

In this paper, we introduce and study shifted twisted quantum affine algebras which provide a twisted counterpart of the theory of shifted quantum affine algebras. The shifted twisted quantum affine algebra $\U_q^{μ_+,μ_-}(\hgs)$ is obtained from the Drinfeld current presentation of twisted quantum loop algebras by shifting the Cartan--Drinfeld currents ${\varphi }_i^{\pm }(z)$ according to a coweight pair $({\mu }_{+},{\mu }_{-})$. We prove that it admits a triangular decomposition and that, up to isomorphism, they depend only on the total shift $\mu ={\mu }_{+}+{\mu }_{-}$. For each shift $\mu$, we define a category ${\mathcal{O}}_{\mu }$ of representations of $\U_q^μ(\hgs) = \U_q^{0,μ}(\hgs)$ and prove a rationality theorem for the Cartan currents: on every weight space, the two currents ${\varphi }_i^{+}(z)$ and ${\varphi }_i^{-}(z)$ are expansions of the same rational operator-valued function, whose degree is prescribed by ${\alpha }_i(\mu )$. As a consequence, we classify the simple objects of ${\mathcal{O}}_{\mu }$ by rational $\ell$-weights of the corresponding degrees. We then construct a deformed Drinfeld coproduct and use it to define a fusion product on the direct sum ${\mathcal{O}}^{sh}$ of the categories ${\mathcal{O}}_{\mu }$. This fusion product is compatible with $q$-characters. We also classify finite-dimensional simple modules in ${\mathcal{O}}^{sh}$ in terms of dominant rational $\ell$-weights, with a separate treatment of type $A_{2n}^{(2)}$. Finally, we construct restriction representations relating representations of twisted quantum affine Borel algebras to representations of shifted twisted quantum affine algebras, and establish a $q$-characters formula for simple finite-dimensional representations of shifted twisted quantum affine algebras in terms of the $q$-characters of the corresponding simple representations of the twisted quantum affine Borel algebra $\U_q(\bs)$.