Categorical crystals for quantum affine algebras

Abstract

Abstract In this paper, a new categorical crystal structure for the quantum affine algebras is presented. We introduce the notion of extended crystals B ^ 𝔤 ⁢ ( ∞ ) {\widehat{B}_{{\mathfrak{g}}}(\infty)} for an arbitrary quantum group U q ⁢ ( 𝔤 ) {U_{q}({\mathfrak{g}})} , which is the product of infinite copies of the crystal B ⁢ ( ∞ ) {B(\infty)} . For a complete duality datum 𝒟 {{\mathcal{D}}} in the Hernandez–Leclerc category 𝒞 𝔤 0 {{\mathscr{C}_{\mathfrak{g}}^{0}}} of a quantum affine algebra U q ′ ⁢ ( 𝔤 ) {U_{q}^{\prime}({\mathfrak{g}})} , we prove that the set ℬ 𝒟 ⁢ ( 𝔤 ) {\mathcal{B}_{{\mathcal{D}}}({\mathfrak{g}})} of the isomorphism classes of simple modules in 𝒞 𝔤 0 {{\mathscr{C}_{\mathfrak{g}}^{0}}} has an extended crystal structure isomorphic to B ^ 𝔤 fin ⁢ ( ∞ ) {\widehat{B}_{{{\mathfrak{g}}_{\mathrm{fin}}}}(\infty)} . An explicit combinatorial description of the extended crystal ℬ 𝒟 ⁢ ( 𝔤 ) {\mathcal{B}_{{\mathcal{D}}}({\mathfrak{g}})} for affine type A n ( 1 ) {A_{n}^{(1)}} is given in terms of affine highest weights.