For a given hereditary abelian category satisfying some finiteness conditions, in certain twisted cases it is shown that the modified Ringel-Hall algebra is isomorphic to the naive lattice algebra and there exists an epimorphism from the modified Ringel-Hall algebra to the lattice algebra. Furthermore, the kernel of this epimorphism is described explicitly. Finally, we show that the naive lattice algebra is invariant under the derived equivalences of hereditary abelian categories.
We use semi-derived Ringel-Hall algebras of quivers with loops to realize the whole quantum Borcherds-Bozec algebras and quantum generalized Kac-Moody algebras.
In this paper, the rational Ringel-Hall algebras for tame quivers are introduced and are identified with the positive part of the quantum extended Kac-Moody algebras. By using the rational Ringel-Hall algebras, we show that the existence of Hall polynomials for tame quiver algebras. The PBW bases are constructed and new classes of perverse sheaves are shown to have strong purity property. These allows us to construct the canonical bases of the positive part of the quantum extended Kac-Moody algebras.
In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig (\cite{Lusztig,Lusztig2}) to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula in \cite{Green} and the restriction functor in \cite{Lusztig2}. By a geometric way to prove the Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework.
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 on a smooth projective curve or on a weighted projective line. Inspired by the works of Bridgeland and Gorsky, we define the semi-derived Ringel-Hall algebra of $\mathcal{A}$, denoted by $\mathcal{S}\mathcal{D}\mathcal{H}_{\mathbb{Z}/2}(\mathcal{A})$, to be the localization of a quotient algebra of the Ringel-Hall algebra of the category of $\mathbb{Z}/2$-gr…
In this paper, we give a recursive formula for the interesting PBW basis $E_{A}$ of composition subalgebras of Ringel-Hall algebras $\fkH_\vartri(n)$ of cyclic quivers after \cite{DengDuXiao2007generic}, and another construction of canonical bases of $\U_v^+(\wih{\fks\fkl}_n)$ from the monomial bases $m^{(A)}$ follow \cite{DuZhaomultiplication}. As an application, we will determined all the canonical basis of $\U_v^+(\wih{\fks\fkl}_2)$ associated with modules of Lowery length $\leqs3$. Finally, we will discuss the relation of canonical bases of Ringel-Hall algebras and those of affine quantum …
In [19], Zheng studied the bounded derived categories of constructible $\bar{\mathbb{Q}}_l$-sheaves on some algebraic stacks consisting of the representations of a enlarged quiver and categorified the integrable highest weight modules of the corresponding quantum group by using these categories. In this paper, we shall generalize Zheng's work to highest weight modules of a subalgebra of the double Ringel-Hall algebra associated to a quiver in a functional version.
In this paper, we study the category $\mathscr{H}^{(ρ)}$ of semi-stable coherent sheaves of a fixed slope $ρ$ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of $\mathscr{H}^{(ρ)}$ and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.
In this paper we define the modified Ringel-Hall algebra $\cm\ch(\ca)$ of a hereditary abelian category $\ca$ from the category $C^b(\mathcal{A})$ of bounded $\mathbb{Z}$-graded complexes. Two main results have been obtained. One is to give a new proof of Green's formula on Ringel-Hall numbers by using the associative multiplication of the modified Ringel-Hall algebra. The other is to show that in certain twisted cases the derived Hall algebra can be embedded in the modified Ringel-Hall algebra. As a consequence of the second result, we get that in certain twisted cases the modified Ringel-Hal…
We give the Ringel-Hall algebra construction of the positive half of quantum Borcherds-Bozec algebras as the generic composition algebras of quivers with loops.
We define and investigate nilpotent Lie algebras associated with quadratic forms. We also present their connections with Lie algebras and Ringel-Hall algebras associated with representation directed algebras.
For any finitely dimensional associative algebra with global dimension $\leq 2$, we show that there is an embedding from the twisted Ringel-Hall algebra to the Brigeland's Ringel-Hall algebra. In particular, this result is true for tilted algebras and canonical algebras.
For an integer $w$, let $\cs_w$ be the algebraic triangulated category generated by a $w$-spherical object. We determine the Picard group of $\cs_w$ and show that each orbit category of $\cs_w$ is triangulated and is triangle equivalent to a certain orbit category of the bounded derived category of a standard tube. When $n=2$, the orbit category $\cs_w/Σ^2$ is 2-periodic triangulated, and we characterize the associated Ringel--Hall Lie algebra in the sense of Peng and Xiao.
In \cite{rupel3},the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category $\mathcal{A}$ to an appropriate $q$-polynomial algebra. In the case that $\mathcal{A}$ is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in \cite{DX}. Moreover, if the underlying graph of $Q$ is bipartite and the matrix $B$ associated to the quiver $Q$ is of full rank, we show that the image of the algebra homomorphisms is in the corresponding quantum cluster algebra.
Based on the realization of quantum Borcherds-Bozec algebra $\widetilde{\mathbf{U}}$ and quantum generalized Kac-Moody algebra ${}^B\widetilde{\mathbf{U}}$ via semi-derived Ringel-Hall algebra of a quiver with loops, we deduce the braid group actions of $\widetilde{\mathbf{U}}$ introduced by Fan and Tong recently and establish braid group actions for ${}^B\widetilde{\mathbf{U}}$ by applying the BGP reflection functors to semi-derived Ringel-Hall algebras.
Let $A$ be a tame hereditary algebra over a finite field $k$ with $q$ elements, and ${\bar{A}}$ be the duplicated algebra of $A$. In this paper, we investigate the structure of Ringel-Hall algebra $\mathscr{H} (\bar{A})$ and of the corresponding composition algebra $\mathscr{C} (\bar{A})$. As an application, we prove the existence of Hall polynomials $g_{XY}^M$ for any $\bar{A}$-modules $M, X$ and $Y$ with $X$ and $Y$ indecomposable if $A$ is a tame quiver $k$-algebra, then we also obtain some Lie subalgebras induced by $\bar{A}$.
We compute the Euler characteristics of quiver Grassmannians and quiver flag varieties of tree and band modules and prove their positivity. This generalizes some results by G.C. Irelli [arXiv:0910.2592]. As an application we consider the Ringel-Hall algebra $C(A)$ of some string algebras $A$ and compute in combinatorial terms the products of arbitrary functions in $C(A)$.
We define two classes of representations of quivers over arbitrary fields, called monomorphic representations and epimorphic representations. We show that every representation has a unique maximal nilpotent subrepresentation and the associated quotient is always monomorphic, and every representation has a unique maximal epimorphic subrepresentation and the associated quotient is always nilpotent. The uniquenesses of such subrepresenations imply two identities in the Ringel-Hall algebra. By applying Reineke's integration map, we obtain two identities in the corresponding quantum torus.
Let $\mathbf{U}$ be the quantized enveloping algebra and $\dot{\mathbf{U}}$ its modified form. Lusztig gives some symmetries on $\mathbf{U}$ and $\dot{\mathbf{U}}$. Since the realization of $\mathbf{U}$ by the reduced Drinfeld double of the Ringel-Hall algebra, one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig's symmetries on $\mathbf{U}$ and their important properties, for instance, the braid relations. In this paper, we define a modified form $\dot{\mathcal{H}}$ of the Ringel-Hall algebra and realize the Lusztig's symmetries on $\dot{\mathbf{U}}$ b…
Let k be an arbitrary field and Q an acyclic quiver of tame type. Consider the path algebra kQ and the category of finite dimensional right modules Mod kQ. In the first part of the paper we deduce that the Gabriel-Roiter inclusions in preprojective indecomposables and simple homogeneous regulars (with dimension the minimal radical vector) as well as their Gabriel-Roiter measures are field independent. Using this result we can prove in a more general setting a theorem by Bo Chen which states that the Gabriel-Roiter submodule of a simple homogeneous regular (with dimension the minimal radical ve…
In this paper, we study the two-parameter quantum group $U_{r,s}(\mathfrak sl_{\infty})$ associated to the Lie algebra $\mathfrak sl_{\infty}$ of infinite rank. We shall prove that the two-parameter quantum group $U_{r,s}(\mathfrak sl_{\infty})$ admits both a Hopf algebra structure and a triangular decomposition. In particular, it can be realized as the Drinfeld double of it's certain Hopf subalgebras. We will also study a two-parameter twisted Ringel-Hall algebra $H_{r,s}(A_{\infty})$ associated to the category of all finite dimensional representations of the infinite linear quiver $A_{\infty…
We show that the reduced Drinfeld double of the Ringel-Hall algebra of a hereditary category is invariant under derived equivalences. By associating an explicit isomorphism to a given derived equivalence, we also extend the results of [BS1], [BS2], [SVdB], and [XY].
We investigate the structure of the double Ringel-Hall algebras associated with cyclic quivers and its connections with quantum loop algebras of $\mathfrak{gl}_n$, affine quantum Schur algebras and affine Hecke algebras. This includes their Drinfeld-Jimbo type presentation, affine quantum Schur-Weyl reciprocity, representations of affine quantum Schur algebras, and connections with various existing works by Lusztig, Varagnolo-Vasserot, Schiffmann, Hubery, Chari-Pressley, Frenkel-Mukhin, etc. We will also discuss conjectures on a realization of Beilinson-Lusztig-MacPherson type and Lusztig type…
Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel--Hall algebra ${\mathscr D}(Q)$ of a quiver $Q$ as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation $L(Λ_0)$ of ${\mathscr D}(Δ_n)$ of the cyclic quiver $Δ_n$ provides a realization of the $q$-deformed Fock space $\bigwedge^\infty$ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of $Δ_n$, we obtain a decomposition of the basic representation $L(Λ_…
We will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra ${\frak H}_Δ(n)$ of a cyclic quiver $Δ(n)$ given in \cite[Thm~4.5]{DuFu2015quantum}. As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established in \cite{Guo1995hallpoly} and \cite{Ringel1993composition}, and derive a recursive formula to compute them. We will further use the formula and the construction of certain monomial base for ${\mathfrak H}_Δ(n)$ given in \cite{DengDuXiao2007generic}, together with the double Ringel--Hall algebra…
A quantum symmetric pair consists of a quantum group $\mathbf U$ and its coideal subalgebra ${\mathbf U}^{\imath}_{\boldsymbolς}$ with parameters $\boldsymbolς$ (called an $\imath$quantum group). We initiate a Hall algebra approach for the categorification of $\imath$quantum groups. A universal $\imath$quantum group $\widetilde{\mathbf U}^{\imath}$ is introduced and ${\mathbf U}^{\imath}_{\boldsymbolς}$ is recovered by a central reduction of $\widetilde{\mathbf U}^{\imath}$. The semi-derived Ringel-Hall algebras of the first author and Peng, which are closely related to semi-derived Hall algeb…
Our investigation in the present paper is based on three important results. (1) In [12], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [4], Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simp…
In this paper, the singular Ringel-Hall algebra for a tame quiver is introduced and shown to be isomorphic to the positive part of the quantum extended Kac-Moody algebra. A PBW basis is constructed and a new class of perverse sheaves is shown to have purity property. This allows to construct the canonical bases of the positive part of the quantum extended Kac-Moody algebra. As an application, the existence of Hall polynomials for tame quiver algebras is proved.
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 in most ways a finitary abelian category, but is not additive. As an application, we define and study the Hall algebra of $Coh_n(\Pone)$. We show that it is isomorphic as a Hopf algebra to the enveloping algebra of the product of a non-standard Borel in the loop algebra $L {\mathfrak{gl}}_2$ and an abelian Lie algebra on infinitely many generators. This shoul…
Inspired by recent work of Bridgeland, from the category C^b(E) of bounded complexes over an exact category E satisfying certain finiteness conditions, we construct an associative unital "semi-derived Hall algebra" SDH(E). This algebra is an object sitting, in some sense, between the usual Hall algebra H(C^b(E)) and the Hall algebra of the bounded derived category D^b(E), introduced by Toen and further generalized by Xiao and Xu. It has the structure of a free module over a suitably defined quantum torus of acyclic complexes, with a basis given by the isomorphism classes of objects in the boun…
Let Q be a quiver. M. Reineke and A. Hubery investigated the connection between the composition monoid, as introduced by M. Reineke, and the generic composition algebra, as introduced by C. M. Ringel, specialised at q=0. In this thesis we continue their work. We show that if Q is a Dynkin quiver or an oriented cycle, then the composition algebra at q=0 is isomorphic to the monoid algebra of the composition monoid. Moreover, if Q is an acyclic, extended Dynkin quiver, we show that there exists an epimorphism from the composition algebra at q=0 to the monoid algebra of the composition monoid, an…
The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field FF_q and any sequence ii of simple objects in C the element X_{V,ii} of the corresponding algebra P_{C,ii} of q-polynomials. We prove that if C was hereditary, then the assignments V-> X_{V,ii} define algebra homomorphisms from the (dual) Hall-Ringel algebra of C to the P_{C,ii}, which generalize the well-known Feigin homomorphisms from the upper half of a quantum group to q-polynomial algebras. If C is the representation cate…
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 $\mathfrak{n}_-$ as above, we provide an infinite collection of cyclic finite abelian subgroups of $SL_3(\mathbb{C})$ with the following properties. Let $G$ be any group in the collection, $Y=G\operatorname{-}\mbox{Hilb}(\mathbb{C}^3)$ and $Ψ: D^b_G(Coh(\mathbb{C}^3))\rightarrow D^b(Coh(Y))$ the derived equivalence of Bridgeland, King and Reid. We present an (expli…
