Related papers: Quantum Algebras and Cyclic Quiver Varieties

In this paper we introduce several computational techniques for the study of moduli spaces of McKay quiver representations, making use of Groebner bases and toric geometry. For a finite abelian group G in GL(n,k), let Y_\theta be the…

Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg)…

We construct a family of PBWD bases for the positive subalgebras of quantum loop algebras of type $C_n$ and $D_n$, as well as their Lusztig and RTT integral forms, in the new Drinfeld realization. We also establish a shuffle algebra…

These notes reflect the contents of three lectures given at the workshop of the 14th International Conference on Representations of Algebras (ICRA XIV), held in August 2010 in Tokyo. We first provide an introduction to quantum loop algebras…

We consider the motivic Hall algebra of coherent sheaves over an irreducible reduced projective curve of arithmetic genus $1$. We introduce the composition subalgebra in the singular curve case, and show that it is isomorphic to the…

For a finite subgroup $\Gamma\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert scheme of $n$ points on the Kleinian singularity $\mathbb{C}^2/\Gamma$ as a Nakajima quiver variety for…

In the recent papers with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux-Leclerc-Thibon-Ariki type conjectures for the affine Hecke algebras of type $B$. Namely, we conjectured that…

We realize geometrically a family of simple modules of (shifted) quantum loop groups including Kirillov-Reshetikhin and prefundamental representations. To do this, we introduce a new family of algebras attached to quivers with potentials,…

This paper is to construct unstable, Morita stable and stable bivariant algebraic Kasparov $K$-theory spectra of $k$-algebras. These are shown to be homotopy invariant, excisive in each variable $K$-theories. We prove that the spectra…

We unify problems about the equivariant geometry of symmetric quiver representation varieties, in the finite type setting, with the corresponding problems for symmetric varieties $GL(n)/K$ where $K$ is an orthogonal or symplectic group. In…

We find new presentations of the modified Ariki-Koike algebra (known also as Shoji's algebra) $\mathcal H_{n,r}$ over an integral domain $R$ associated with a set of parameters $q,u_1,\ldots,u_r$ in $R$. It turns out that the algebra…

We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra $\mathscr{S}_{\Sigma,\mathbb{W}}^{A}$ of a walled surface $(\Sigma,\mathbb{W})$, and prove that it has a quantum…

We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra $H$ endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups…

We first study a new family of graded quiver varieties together with a new $t$-deformation of the associated Grothendieck rings. This provides the geometric foundations for a joint paper by Yoshiyuki Kimura and the author. We further…

We show that the shuffle algebras for polylogarithms and regularized MZVs in the sense of Ihara, Kaneko and Zagier are both free commutative nonunitary Rota-Baxter algebras with one generator. We apply these results to show that the full…

Let $Q$ be an arbitrary finite quiver. We use nonabelian stable envelopes to relate representations of the Maulik-Okounkov Lie algebra $\mathfrak{g}^{MO}_Q$ to representations of the BPS Lie algebra associated to the tripled quiver $\tilde…

Buan, Iyama, Reiten and Smith proved that cluster-tilting objects in triangulated 2-Calabi--Yau categories are closely connected with mutation of quivers with potentials over an algebraically closed field. We prove a more general statement…

The Ohno-Nakagawa (O-N) reflection theorem is an unexpectedly simple identity relating the number of $\mathrm{GL}_2 \mathbb{Z}$-classes of binary cubic forms (equivalently, cubic rings) of two different discriminants $D$, $-27D$; it…

To a quiver $Q$ and choices of nonzero scalars $q_i$, non-negative integers $\alpha_i$, and integers $\theta_i$ labeling each vertex $i$, Crawley-Boevey--Shaw associate a "multiplicative quiver variety" $\mathcal{M}_\theta^q(\alpha)$, a…

Automorphisms of algebras $R$ from a very large axiomatic class of quantum nilpotent algebras are studied using techniques from noncommutative unique factorization domains and quantum cluster algebras. First, the Nakayama automorphism of…