Crystal Bases for Quantum Generalized Kac-Moody Algebras

Let ${\mathbf U}^-_q$ be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the fixed point subalgebra of ${\mathfrak g}$ obtained from a diagram automorphism $\sigma$ on ${\mathfrak g}$. Let ${\mathbf B}^{\sigma}$ be the set of $\sigma$-fixed elements in the canonical basis of ${\mathbf U}_q^-$, and $\underline{\mathbf B}$ the canonical basis of $...

Let $\mathcal{O}^{int}_q(m|n)$ be a semisimple tensor category of modules over a quantum ortho-symplectic superalgebra of type $B, C, D$ introduced in the author's previous work. It is a natural counterpart of the category of finitely dominated integrable modules over a quantum group of type $B, C, D$ from a viewpoint of super duality. Continuing the previous work on type $B$ and $C$, we classify the irreducible modules in $\mathcal{O}^{int}_q(m|n)$, and prove the existence a...

We provide some necessary details to several arguments appearing in our previous paper ``Canonical bases for quantum generalized Kac-Moody algebras''. We also make the link with some other work on the same subject.

We consider imaginary Verma modules for quantum affine algebraU_q(\widehat{\mathfrak{sl}(2)}) and define a crystal-like base which we call an imaginary crystal basis using the Kashiwara algebra K_q constructed in earlier work of the authors. In particular, we prove the existence of imaginary like bases for a suitable category of reduced imaginary Verma modules for U_q(\widehat{\mathfrak{sl}(2)}).

We study the crystal structure on categories of graded modules over algebras which categorify the negative half of the quantum Kac-Moody algebra associated to a symmetrizable Cartan data. We identify this crystal with Kashiwara's crystal for the corresponding negative half of the quantum Kac-Moody algebra. As a consequence, we show the simple graded modules for certain cyclotomic quotients carry the structure of highest weight crystals, and hence compute the rank of the corre...

The Kirillov--Reshetikhin modules W^{r,s} are finite-dimensional representations of quantum affine algebras U'_q(g), labeled by a Dynkin node r of the affine Kac--Moody algebra g and a positive integer s. In this paper we study the combinatorial structure of the crystal basis B^{2,s} corresponding to W^{2,s} for the algebra of type D_n^{(1)}.

The generalized quantum group of type $A$ is an affine analogue of quantum group associated to a general linear Lie superalgebra, which appears in the study of solutions to the tetrahedron equation or the three-dimensional Yang-Baxter equation. In this paper, we develop the crystal base theory for finite-dimensional representations of generalized quantum group of type $A$. As a main result, we construct Kirillov-Reshetikhin modules, that is, a family of irreducible modules wh...

A new categorical crystal structure for the quantum affine algebras is presented. We introduce the extended crystal $\widehat{B}_{\mathfrak{g}}(\infty)$ for an arbitrary quantum group, which is the product of infinite copies of the crystal $B(\infty)$. For a complete duality datum in the Hernandez-Leclerc category $\mathcal{C}^0_{\mathfrak{g}}$ of a quantum affine algebra $U_q'(\mathfrak{g})$, we prove that the set of the isomorphism classes of simple modules in $\mathcal{C}^...

Let B(\infty) be the crystal corresponding to the nilpotent part of a quantized Kac-Moody algebra. We suggest a general way to represent B(\infty) as the set of integer solutions of a system of linear inequalities. As an application, we treat in a unified manner all Kac-Moody algebras of rank 2 (sharpening the result by Kashiwara), as well as the algebras of types A_n and A_{n-1}^{(1)}.

Polyhedral realization of crystal bases is one of the methods for describing the crystal base $B(\infty)$ explicitly. This method can be applied to symmetrizable Kac-Moody types. We can also apply this method to the crystal bases $B(\lambda)$ of integrable highest weight modules and of modified quantum algebras. But, the explicit forms of the polyhedral realizations of crystal bases $B(\infty)$ and $B(\lambda)$ are only given in the case of arbitrary rank 2, of $A_n$ and of $...