Related papers: A Drinfeld type presentation of affine $\imath$qua…
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.
We put forward the exact version of two-parameter generating functions with $\tau$-invariance, which allows us to give a unified and inherent definition for the Drinfeld realization of two-parameter quantum affine algebras for all the…
In this paper, we extend the Drinfeld current realization of quantum affine algebras $U_q(\hat {\gg})$ and of the Yangians in several directions: we construct current operators for non-simple roots of ${\gg}$, define a new braid group…
We construct finite $R$-matrices for the first fundamental representation $V$ of two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ for classical $\mathfrak{g}$, both through the decomposition of $V\otimes V$ into irreducibles…
The $\imath$quantum groups have two realizations: one via the $\imath$Hall algebras and the other via the quantum Grothendieck rings of quiver varieties, as developed by the first author and Wang. The isoclasses of perverse sheaves provide…
We build representations of the elliptic braid group from the data of a quantum D-module M over a ribbon Hopf algebra U. The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid, and also certain…
We establish the $Q \widetilde{Q}$-systems for the twisted quantum affine algebras that were conjectured in arXiv:1606.05301. We develop the representation theory of Borel subalgebra of twisted quantum affine algebras and we construct their…
We introduce a minimalistic presentation for the twisted Yangian ${}^\imath\mathscr Y$ associated with split symmetric pairs (or Satake diagrams) introduced in arXiv:2406.05067 via a Drinfeld type presentation. As applications, we establish…
When the parameter $q\in\mathbb C^*$ is not a root of unity, simple modules of affine $q$-Schur algebras have been classified in terms of Frenkel--Mukhin's dominant Drinfeld polynomials (\cite[4.6.8]{DDF}). We compute these Drinfeld…
We propose a novel quantum integrable model for every non-simply laced simple Lie algebra ${\mathfrak g}$, which we call the folded integrable model. Its spectra correspond to solutions of the Bethe Ansatz equations obtained by folding the…
We study the quantum invariants of projective varieties over the number fields. Namely, explicit formulas for a functor $\mathscr{Q}$ on such varieties are proved. The case of abelian varieties with complex multiplication is treated in…
We study the structure and representation theory of affine wreath product algebras and their cyclotomic quotients. These algebras, which appear naturally in Heisenberg categorification, simultaneously unify and generalize many important…
We compare the reduced Drinfeld doubles of the composition subalgebras of the category of representations of the Kronecker quiver $\overr{Q}$ and of the category of coherent sheaves on ${\mathbb P}^1$. Using this approach, we show that the…
We give a uniform, Lie-theoretic mirror symmetry construction for the Frobenius manifolds defined by Dubrovin-Zhang in arXiv:hep-th/9611200 on the orbit spaces of extended affine Weyl groups, including exceptional Dynkin types. The B-model…
We construct a version of Dijkgraaf-Witten theory based on a compact abelian Lie group within the formalism of Turaev's homotopy quantum field theory. As an application we show that the 2+1-dimensional theory based on U(1) classifies lens…
Recently, a birational representation of an extended affine Weyl group of $(A_{2N}\rtimes A_1)^{(1)}$-type, which gives a higher-order generalization of an $A_4^{(1)}$-surface type $q$-Painlev\'e equation, was obtained. In this paper, we…
We define a multiple Dirichlet series whose group of functional equations is the Weyl group of the affine Kac-Moody root system $\tilde{A}_n$, generalizing the theory of multiple Dirichlet series for finite Weyl groups. The construction is…
Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the…
We express the comultiplication of the generators in Drinfelds second realization of the quantum affine algebra U_q(sl_2^), induced by the comultiplication of the generators in the Drinfeld-Jimbo realization of U_q(sl_2^) in terms of…
A quantum symmetric pair consists of a quantum group $\widetilde{\mathbf{U}}$ and its coideal subalgebra $\widetilde{\mathbf{U}}^\imath$. The Hall algebra constructions of $\widetilde{\mathbf{U}}$ and $\widetilde{\mathbf{U}}^\imath$ are…