Related papers: $\imath$Hall algebra of the projective line and $q…

The $\imath$Hall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that…

We survey some recent development on the theory of $\imath$Hall algebras. Starting from $\imath$quivers (aka quivers with involutions), we construct a class of 1-Gorenstein algebras called $\imath$quiver algebras, whose semi-derived Hall…

From a category $\mathcal{A}$ with an involution $\varrho$, we introduce $\varrho$-complexes, which are a generalization of (bounded) complexes, periodic complexes and modules of $\imath$quiver algebras. The homological properties of the…

We establish a Drinfeld type new presentation for the $\imath$quantum groups arising from quantum symmetric pairs of split affine ADE type, which includes the $q$-Onsager algebra as the rank 1 case. This presentation takes a form which can…

We show that the morphism $\Omega$ from the $\imath$quantum loop algebra $^{\texttt{Dr}}\widetilde{\mathbf{U}}(L\mathfrak{g})$ of split type to the $\imath$Hall algebra of the weighted projective line is injective if $\mathfrak{g}$ is of…

To an abelian category A of homological dimension 1 satisfying certain finiteness conditions, one can associate an algebra, called the Hall algebra. Kapranov studied this algebra when A is the category of coherent sheaves over a smooth…

The $\imath$quiver algebras were introduced recently by the authors to provide a Hall algebra realization of universal $\imath$quantum groups, which is a generalization of Bridgeland's Hall algebra construction for (Drinfeld doubles of)…

A quantum symmetric pair consists of a quantum group $\mathbf U$ and its coideal subalgebra ${\mathbf U}^{\imath}_{\boldsymbol{\varsigma}}$ with parameters $\boldsymbol{\varsigma}$ (called an $\imath$quantum group). We initiate a Hall…

Let $\mathcal{A}$ be an arbitrary hereditary abelian category that may not have enough projective objects. For example, $\mathcal{A}$ can be the category of finite-dimensional representations of a quiver or the category of coherent sheaves…

For an essentially small hereditary abelian category $\mathcal{A}$, we define a new kind of algebra $\mathcal{H}_{\Delta}(\mathcal{A})$, called the $\Delta$-Hall algebra of $\mathcal{A}$. The basis of $\mathcal{H}_{\Delta}(\mathcal{A})$ is…

Recently the authors initiated an $\imath$Hall algebra approach to (universal) $\imath$quantum groups arising from quantum symmetric pairs. In this paper we construct and study BGP type reflection functors which lead to isomorphisms of the…

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…

Let X be a smooth projectibe curve over a finite field. We consider the Hall algebra H whose basis is formed by isomorphism classes of coherent sheaves on X and whose typical structure constant is the number of subsheaves in a given sheaf…

We generalize a theorem of Kapranov by showing that the Hall algebra of the category of coherent sheaves on a weighted projective line (over a finite field) provides a realization of the (quantized) enveloping algebra of a certain nilpotent…

Let Q be a finite quiver without sources, and A be the corresponding algebra with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of projective A-modules. We call such a…

A Laurent polynomial ring $A[t,1/t]$ with coefficients in a unital ring $A$ determines a category of quasi-coherent sheaves on the projective line over $A$; its $K$-theory is known to split into a direct sum of two copies of the $K$-theory…

We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined by Licata and Savage. We also show that as an algebra, it is isomorphic to "half" of a central extension of the elliptic Hall…

We describe recent work on preprojective algebras and moduli spaces of their representations. We give an analogue of Kac's Theorem, characterizing the dimension types of indecomposable coherent sheaves over weighted projective lines in…

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…

The quantum loop algebra $U_{v}(\mathcal{L}\mathfrak{g})$ was defined as a generalization of the Drinfeld's new realization of the quantum affine algebra to the loop algebra of any Kac-Moody algebra $\mathfrak{g}$. It has been shown by…