Related papers: Noncommutative projective curves and quantum loop …
In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category…
In \cite{FT19}, Finkelberg and Tsymbaliuk introduced the notion of shifted quantum affine algebras and described their role in the study of quantized Coulomb branches associated to certain 3D $N = 4$ quiver gauge theories. We describe a new…
In this paper, we define the $K$-theoretic Hall algebra for $0$-dimensional coherent sheaves on a smooth projective surface, prove that the algebra is associative and construct a homomorphism to a redefined shuffle algebra analogous to…
By using cocycle deformation, we construct a certain class of Hopf algebras, containing the quantized enveloping algebras and their analogues, from what we call pre-Nichols algebras. Our construction generalizes in some sense the known…
We define and study the category $Coh_n(\Pone)$ of normal coherent sheaves on the monoid scheme $\Pone$ (equivalently, the $\mathfrak{M}_0$-scheme $\Pone / \fun$ in the sense of Connes-Consani-Marcolli \cite{CCM}). This category resembles…
We develop the approach via quasihomomorphisms and the universal algebra $qA$ to Kasparov's $KK$-theory, so as to cover versions of $KK$ such as $KK^{nuc}$, $KK^G$ and ideal related $KK$-theory.
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…
Let $\mathfrak{g}\neq \mathfrak{so}_8$ be a simple Lie algebra of type $A,D,E$ with $\widehat{\mathfrak{g}}$ the corresponding affine Kac-Moody algebra and $\mathfrak{n}_-\subset \widehat{\mathfrak{g}}$ a nilpotent subalgebra. Given…
We continue our study of noncommutative resolutions of Coulomb branches in the case of quiver gauge theories. These include the Slodowy slices in type A and symmetric powers in $\mathbb{C}^2$ as special cases. These resolutions are based on…
By using perverse sheaves on representation spaces of quivers over $k[t]/(t^n)$ and jet schemes over flag varieties, we construct a geometric composition algebra $\mathbf K$ under Lusztig's framework on geometric realizations of the…
Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic…
We show that the Hall algebra of the category of coherent sheaves on an elliptic curve (or, equivalently, the algebra of unramified automorphic forms for GL(n) for all n) is equal to the stable limit of spherical double affine Hecke…
We generalize Kuznetsov's theory of homological projective duality to the setting of noncommutative algebraic geometry. Simultaneously, we develop the theory over general base schemes, and remove the usual smoothness, properness, and…
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…
Coideal subalgebras of the quantized enveloping algebra are surveyed, with selected proofs included. The first half of the paper studies generators, Harish-Chandra modules, and associated quantum homogeneous spaces. The second half…
Let $A$ be the path algebra of a finite acyclic quiver $Q$ over a finite field. We realize the quantum cluster algebra with principal coefficients associated to $Q$ as a sub-quotient of a certain Hall algebra involving the category of…
We formalize the quantum arithmetic, i.e. a relationship between number theory and operator algebras. Namely, it is proved that rational projective varieties are dual to the $C^*$-algebras with real multiplication. Our construction fits all…
We extend a theorem, originally formulated by Blattner-Cohen-Montgomery for crossed products arising from Hopf algebras weakly acting on noncommutative algebras, to the realm of left Hopf algebroids. Our main motivation is an application to…
We describe certain quiver Hopf algebras by parameters. This leads to the classification of multiple Taft algebras as well as pointed Yetter-Drinfeld modules and their corresponding Nichols algebras. In particular, when the ground-field $k$…
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…