Related papers: Abstract crystals for quantum Borcherds-Bozec alge…

Let $G$ be a split connected reductive group over a non-archimedan local field $F$. The depth zero stable Bernstein conjecture asserts that there is an algebra isomorphism between the depth zero stable Bernstein center of $G(F)$ and the…

We give a realization of the level zero fundamental weight representation $W(\varpi_k)$ of the quantum affine algebra $U_q'(\mf{g})$, when $\mf{g}$ has a maximal parabolic subalgebra of type $C_n$. We define a semisimple $U'_q({\mf…

We show that in case a cluster algebra coincides with its upper cluster algebra and the cluster algebra admits a grading with finite dimensional homogeneous components, the corresponding Berenstein-Zelevinsky quantum cluster algebra can be…

We categorify a class of quantum groups associated with quivers, possibly with loops, by constructing the corresponding Khovanov-Lauda-Rouquier algebras (KLR) algebras $R$. We prove that the indecomposable projective $R$-modules realize the…

We present a uniform construction of level 1 perfect crystals $\mathcal B$ for all affine Lie algebras. We also introduce the notion of a crystal algebra and give an explicit description of its multiplication. This allows us to determine…

We show that the different labelings of the crystal graph for irreducible highest weight $\mathcal{U}\_q (\hat{\mathfrak{sl}}\_e)$-modules lead to different labelings of the simple modules for non semisimple Ariki-Koike algebras by using…

We introduce a $p$-adic analytic analogue of Backelin and Kremnizer's construction of the quantum flag variety of a semisimple algebraic group, when $q$ is not a root of unity and $| q-1|<1$. We then define a category of $\lambda$-twisted…

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…

This is a continuation of a previous study of quantum vertex algebras of Zamolodchikov-Faddeev type. In this paper, we focus our attention on the special case associated to diagonal unitary rational quantum Yang-Baxter operators. We prove…

Crystal base of the level 0 part of the modified quantum affine algebra $\widetilde U_q(\widehat{sl_2})_0$ is given by path. Weyl group actions, extremal vectors and crystal structure of all irreducible components are described explicitly.

A combinatorial description of the crystal $\mathcal{B}(\infty)$ for finite-dimensional simple Lie algebras in terms of Young tableaux was developed by J. Hong and H. Lee. Using this description, we obtain a combinatorial rule for…

Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite dimensional simply-laced Lie algebras.…

We discuss Cauchy type decompositions of crystal graphs for general linear Lie superalgebras. More precisely, we consider bicrystal graph structures on various sets of matrices of non-negative integers, and obtain their decompositions with…

Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac-Moody algebra. The underlying set consists of the irreducible components of Lusztig's quiver varieties, which are varieties of nilpotent…

Given a state on an algebra of bounded quantum-mechanical observables (the self-adjoint part of a C*-algebra), we investigate those subalgebras that are maximal with respect to the property that the given state's restriction to the…

In this paper we construct a new family of representations for the quantum toroidal algebras of type $A_n$, which are $\ell$-extremal in the sense of Hernandez [24]. We construct extremal loop weight modules associated to level 0…

A class of highest weight irreducible representations of the algebra $U_h(A_\infty)$, the quantum analogue of the completion and central extension $A_\infty$ of the Lie algebra $gl_\infty$, is constructed. It is considerably larger than the…

We give a brief survey of recent developments in the highest weight representation theory and the crystal basis theory of the quantum queer superalgebra $U_q(\mathfrak{q}(n))$.

We prove a highest weight theorem classifying irerducible finite--dimensional representations of quantum affine algebras and survey what is currently known about the structure of these representations.

Let ${\mathbf U}^-_q$ be the negative part of the quantized enveloping algebra associated to a Kac-Moody algebra ${\mathfrak g}$ of symmetric type, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the orbit algebra ${\mathfrak…