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Let $G$ be a simple, simply connected, simply laced algebraic group. We construct a monoidal category of representations of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ whose Grothendieck ring contains a cluster algebra with…
In \cite{CK2005} and \cite{SZ}, the authors constructed the bases of cluster algebras of finite types and of type $\widetilde{A}_{1,1}$, respectively. In this paper, we will deduce $\mathbb{Z}$-bases for cluster algebras of affine types.
Let $O_D$ be the ring of integers in a division algebra of invariant $1/n$ over a p-adic local field. Drinfeld proved that the moduli problem of special formal $O_D$-modules is representable by Deligne's formal scheme version of the…
This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection.…
The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with gl(N). We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the…
There is a well-known presentation for finite and affine Weyl groups called the {\it presentation by conjugation}. Recently, it has been proved that this presentation holds for certain sub-classes of extended affine Weyl groups, the Weyl…
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous…
We study representations of the double affine Lie algebra associated to a simple Lie algebra. We construct a family of indecomposable integrable representations and identify their irreducible quotients. We also give a condition for the…
The quantum group version of the Bernstein-Gelfand-Gelfand resolution is used to construct a double complex of U_q(g)-modules with exact rows and columns. The locally finite dual of its total complex is identified with the de Rham complex…
The study of certain union $X(\mu, b)$ of affine Deligne-Lusztig varieties in the affine flag varieties arose from the study of Shimura varieties with Iwahori level structure. In this paper, we give an explicit formula of $\dim X(\mu, b)$…
We study products of the affine geometric crystal of type A corresponding to symmetric powers of the standard representation. The quotient of this product by the R-matrix action is constructed inside the unipotent loop group. This quotient…
To each symmetric algebra we associate a family of algebras that we call quantum affine wreath algebras. These can be viewed both as symmetric algebra deformations of affine Hecke algebras of type $A$ and as quantum deformations of affine…
We use the idea of free fields to obtain highest weight representations for the extended affine Lie algebra $\widetilde{\frak{gl}_{2}(\bc_q)}$ coordinatized by the quantum torus $\bc_q$ and go on to construct a contravariant hermitian form.…
We show that, under Dubrovin's notion of ''almost'' duality, the Frobenius manifold structure on the orbit spaces of the extended affine Weyl groups of type $\mathrm{ADE}$ is dual, for suitable choices of weight markings, to the equivariant…
Let g be a complex semisimple Lie algebra, tau a point in the upper half-plane, and h a complex deformation parameter such that the image of h in the elliptic curve E_tau is of infinite order. In this paper, we give an intrinsic definition…
This paper studies affine Deligne-Lusztig varieties $X_{\tw}(b)$ in the affine flag variety of a quasi-split tamely ramified group. We describe the geometric structure of $X_{\tw}(b)$ for a minimal length element $\tw$ in the conjugacy…
A massless scalar field in two spacetime dimensions splits into two independent sectors of left and right-moving modes on the light cone. At the quantum level, these two sectors carry a representation of the group of affine transformations…
This note is based on my talk at ICCM 2013, Taipei. We give an exposition of the group-theoretic method and recent results on the questions of non-emptiness and dimension of affine Deligne-Lusztig varieties in affine flag varieties.
We introduce and investigate new invariants on the pair of modules $M$ and $N$ over quantum affine algebras $U_q'(\mathfrak{g})$ by analyzing their associated R-matrices. From new invariants, we provide a criterion for a monoidal category…
We present a complete classification for the epimorphisms from the affine Artin-Tits groups $A(\widetilde{A}_{n})$ to the corresponding Coxeter groups $W(\widetilde{A}_{n})$, for $n\geq 1$.