相关论文: Mixed quantum skew Howe duality and link invariant…

In this paper, we contribute to the broad aim of relating invariants of additive and monoidal categorifications of cluster algebras. Specifically, in the setting of representations of a quantum affine algebra $U_q'(\mathfrak{g})$,…

We show that quantum Schur-Weyl duality leads to Markov duality for a variety of asymmetric interacting particle systems. In particular, we consider three cases: (1) Using a Schur-Weyl duality between a two-parameter quantum group and a…

This article investigates the Hodge theory of the moduli space of genus $g$ curves with $n$ marked points, establishing new connections between Schur-Weyl duality for $\mathfrak{sp}_{g}$ and higher Abel-Jacobi invariants. We develop a…

In \cite{fl}, the authors get a new presentation of two-parameter quantum algebra $U_{v,t}(\mathfrak{g})$. Their presentation can cover all Kac-Moody cases. In this paper, we construct a suitable Hopf pairing such that $U_{v,t}(sl_{n})$ can…

Let $R$ be a commutative ring with one and $q$ an invertible element of $R$. The (specialized) quantum group ${\mathbf U} = U_q(\mathfrak{gl}_n)$ over $R$ of the general linear group acts on mixed tensor space $V^{\otimes r}\otimes…

We present a systematic recipe for generating and classifying duality transformations in one-dimensional quantum lattice systems. Our construction emphasizes the role of global symmetries, including those described by (non)-abelian groups…

In this paper, which is mostly a research announcement, we give a new algebraic construction of knot contact homology in the sense of L. Ng [Ng05a]. For a link $L$ in $ {\mathbb R}^3 $, we define a differential graded (DG) $k$-category $…

Given a spin rational homology sphere $Y$ equipped with a $\mathbb{Z}/m$-action preserving the spin structure, we use the Seiberg--Witten equations to define equivariant refinements of the invariant $\kappa(Y)$ from \cite{Man14}, which take…

Skew monoidal categories are monoidal categories with non-invertible `coherence' morphisms. As shown in a previous paper bialgebroids over a ring R can be characterized as the closed skew monoidal structures on the category Mod R in which…

Let $\pi$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension ${\sf cd}_G(\pi)$, the equivariant geometric dimension ${\sf gd}_G(\pi)$, and the equivariant…

The modular data of a modular category $\mathcal{C}$, consisting of the $S$-matrix and the $T$-matrix, is known to be an incomplete invariant of $\mathcal{C}$. More generally, the invariants of framed links and knots defined by a modular…

We define, for a regular scheme $S$ and a given field of characteristic zero $\KK$, the notion of $\KK$-linear mixed Weil cohomology on smooth $S$-schemes by a simple set of properties, mainly: Nisnevich descent, homotopy invariance,…

We develop a geometric approach toward an interplay between a pair of quantum Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's stabilization procedure in the setting of partial flag varieties of type A (resp.…

Let $\mathbb{S}$ denote the oscillatory module over the complex symplectic Lie algebra $\mathfrak{g}= \mathfrak{sp}(\mathbb{V}^{\mathbb{C}},\omega).$ Consider the $\mathfrak{g}$-module…

The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal $R$-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum…

We define a torus $U \subset T = (\mathbb{C}^\times)^K$ which acts on the $\Delta$-Springer varieties $Y_{n,\lambda,s}$ defined by Griffin-Levinson-Woo and give a Borel-style presentation for the equivariant cohomology ring…

It is shown how a chiral Wess-Zumino-Witten theory with globally defined vertex operators and a one-to-one correspondence between fields and states can be constructed. The Hilbert space of this theory is the direct sum of tensor products of…

The quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of G/B. We enhance a result of Fulton and Woodward by showing that the…

Using a geometric argument building on our new theory of graded sheaves, we compute the categorical trace and Drinfel'd center of the (graded) finite Hecke category $\mathsf{H}_W^\mathsf{gr} = \mathsf{Ch}^b(\mathsf{SBim}_W)$ in terms of the…

This paper is the first of a pair that aims to classify a large number of the type $II$ quantum subgroups of the categories $\mathcal{C}(\mathfrak{sl}_{r+1},k)$. In this work we classify the braided auto-equivalences of the categories of…