Cyclic Quiver Research Articles

Remarks on PBW bases of Ringel–Hall algebras of cyclic quivers

In this paper, we give a recursive formula for the interesting PBW basis [Formula: see text] of composition subalgebras [Formula: see text] of Ringel–Hall algebras [Formula: see text] of cyclic quivers after [Generic extensions and canonical bases for cyclic quivers, Canad. J. Math. 59(6) (2007) 1260–1283], and another construction of canonical bases of [Formula: see text] from the monomial bases [Formula: see text] following [Multiplication formulas and canonical basis for quantum affine, [Formula: see text], Canad. J. Math. 70(4) (2018) 773–803]. As an application, we will determine all the canonical basis elements of [Formula: see text] associated with modules of Loewy length [Formula: see text]. Finally, we will discuss the canonical bases between Ringel–Hall algebras and affine quantum Schur algebras.

Spin versions of the complex trigonometric Ruijsenaars–Schneider model from cyclic quivers

AbstractWe study multiplicative quiver varieties associated to specific extensions of cyclic quivers with $m\geq 2$ vertices. Their global Poisson structure is characterized by quasi-Hamiltonian algebras related to these quivers, which were studied by Van den Bergh for an arbitrary quiver. We show that the spaces are generically isomorphic to the case $m=1$ corresponding to an extended Jordan quiver. This provides a set of local coordinates, which we use to interpret integrable systems as spin variants of the trigonometric Ruijsenaars–Schneider (RS) system. This generalizes to new spin cases recent works on classical integrable systems in the RS family.

Categorical representations and KLR algebras

We prove that the KLR algebra associated with the cyclic quiver of length $e$ is a subquotient of the KLR algebra associated with the cyclic quiver of length $e+1$. We also give a geometric interpretation of this fact. This result has an important application in the theory of categorical representations. We prove that a category with an action of $\widetilde{\mathfrak{sl}}_{e+1}$ contains a subcategory with an action of $\widetilde{\mathfrak{sl}}_{e}$. We also give generalizations of these results to more general quivers and Lie types.

Generalized quiver varieties and triangulated categories

In this paper, we introduce generalized quiver varieties which include as special cases classical and cyclic quiver varieties. The geometry of generalized quiver varieties is governed by a finitely generated algebra $${{\mathcal {P}}}$$ : the algebra $${{\mathcal {P}}}$$ is self-injective if the quiver Q is of Dynkin type, and coincides with the preprojective algebra in the case of classical quiver varieties. We show that in the Dynkin case the strata of generalized quiver varieties are in bijection with the isomorphism classes of objects in $${\text {proj}}\,{{\mathcal {P}}}$$ , and that their degeneration order coincides with the Jensen–Su–Zimmermann’s degeneration order on the triangulated category $${\text {proj}}\,{{\mathcal {P}}}$$ . Furthermore, we prove that classical quiver varieties of type $$A_n$$ can be realized as moduli spaces of representations of an algebra $${{\mathcal {S}}}$$ .

Multiplication Formulas and Canonical Bases for Quantum Affine gln

AbstractWe will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra of a cyclic quiver Δ(n). As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established by J. Y. Guo and C. M. Ringel, and derive a recursive formula to compute them. We will further use the formula and the construction of a monomial basis for given by Deng, Du, and Xiao together with the double Ringel-Hall algebra realisation of the quantum loop algebra given by Deng, Du, and Fu to develop some algorithms and to compute the canonical basis for . As examples, we will show explicitly the part of the canonical basis associated with modules of Lowey length at most 2 for the quantum group .

Reflection Functor in the Representation Theory of Preprojective Algebras for Quivers and Integrable Systems

A brief overview of the representation theory of quivers and the associated (deformed) preprojective algebras, as well as of the theories of moduli spaces of these algebras, quiver varieties and a reflection functor, is given. It is proven that a bijection between moduli spaces (in particular, between quiver varieties), which is induced by a reflection function, is the isomorphism of symplectic affine varieties. The Hamiltonian systems on quiver varieties are defined, and the application of a reflection functor to them is described. The review of [1], concerning the case of a cyclic quiver is given, and a role of the reflection functor in this case is clarified. The “spin” integrable generalizations of Calogero–Moser systems and their application to the KP hierarchy generalizations are described.

Kostka-Shoji Polynomials and Lusztig's Convolution Diagram

We propose an $r$-variable version of Kostka-Shoji polynomials $K^-_{\lambda\mu}$ for $r$-multipartitions $\lambda,\mu$. Our version has positive integral coefficients and encodes the graded multiplicities in the space of global sections of a line bundle over Lusztig's iterated convolution diagram for the cyclic quiver $\tilde{A}_{r-1}$.

Spin Calogero–Moser systems for the cyclic quiver

The method of construction of classical integrable systems on quiver varieties is considered in the case of cyclic quiver. For two types of framing we obtained integrable spin generalisations of the Calogero–Moser systems associated with the complex reflection groups Sn ⋉ (ℤ/mℤ)n. This method gives Lax matrices of the systems in explicit form. In some particular cases the first Hamiltonians of the system are written explicitly. The paper is based on the work of the author with Oleg Chalykh (the University of Leeds).

Stability conditions for cyclic quivers

ABSTRACTLet Δn be a cyclic quiver of n vertices and let 𝒯n denote the category of nilpotent representations of Δn over a field. In the present paper, we study the stability conditions on 𝒯n and describe the semistable subcategories of 𝒯n with a fixed slope. This provides an alternative characterization of the thick subcategories of 𝒯n. Further, we obtain a relationship between the Harder–Narasimhan system for Δn and the Ringel–Hall algebra of Δn.

Multiplicative quiver varieties and generalised Ruijsenaars–Schneider models

We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with $m$ vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction from the space of representations of the quiver. Three families of Poisson-commuting functions are constructed and written explicitly in suitable Darboux coordinates. The case $m=1$ corresponds to the tadpole quiver and the Ruijsenaars-Schneider system and its variants, while for $m>1$ we obtain new integrable systems that generalise the Ruijsenaars-Schneider system. These systems and their quantum versions also appeared recently in the context of supersymmetric gauge theory and cyclotomic DAHAs, as well as in the context of the Macdonald theory.

KP hierarchy for the cyclic quiver

We introduce a generalisation of the KP hierarchy, closely related to the cyclic quiver and the Cherednik algebra Hk(Zm). This hierarchy depends on m parameters (one of which can be eliminated), with the usual KP hierarchy corresponding to the m = 1 case. Generalising the result of Wilson [Invent. Math. 133(1), 1–41 (1998)], we show that our hierarchy admits solutions parameterised by suitable quiver varieties. The pole dynamics for these solutions is shown to be governed by the classical Calogero–Moser system for the wreath-product Zm≀Sn and its new spin version. These results are further extended to the case of the multi-component hierarchy.

Cell Decompositions of Quiver Flag Varieties for Nilpotent Representations of the Cyclic Quiver

Generalizing Schubert cells in type A and a cell decomposition of Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of a cyclic quiver admit an affine cell decomposition parametrized by multi-tableaux. We show that they carry a torus operation with finitely many fixpoints. As an application of the cell decomposition we obtain a vector space basis of certain modules (for quiver Hecke algebras of nilpotent representations of this quiver), similar modules have been studied by Kato as analogues of standard modules.

An elementary construction of monomial bases of modified quantum affine gln

AbstractIn 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum and its canonical basis. A key step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic construction of the canonical basis for modified quantum affine , which among other results used an earlier construction of monomial bases using Ringel–Hall algebra of the cyclic quiver. In this paper, we give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine .

Categorical action of the extended braid group of affine type A

Using a quiver algebra of a double cyclic quiver, we construct a faithful categorical action of the extended braid group of affine type [Formula: see text] on its bounded homotopy category of finitely generated projective modules. The algebra is trigraded and we identify the trigraded dimensions of the space of morphisms of this category with intersection numbers coming from the topological origin of the group.

Cyclotomic Yokonuma–Hecke algebras are cyclotomic quiver Hecke algebras

We prove that cyclotomic Yokonuma–Hecke algebras of type A are cyclotomic quiver Hecke algebras and we give an explicit isomorphism with its inverse, using a similar result of Brundan and Kleshchev on cyclotomic Hecke algebras. The quiver we use is given by disjoint copies of cyclic quivers. We relate this work to an isomorphism of Lusztig.

Quantum dilogarithm identities and cyclic quivers

We study quantum dilogarithm identities for cyclic quivers following Reinekes idea via Ringel-Hall algebra approach. For any given discrete stability function for the cyclic quiver $\Delta_n$ with $n$ vertices, we obtain certain cyclic quantum dilogarithm identities of order $n$ in the sense of Bytsko and Volkov.

Hall algebras of cyclic quivers and q-deformed Fock spaces

Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel–Hall algebra 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(Λ0) of D(Δn) of the cyclic quiver Δn provides a realization of the q-deformed Fock space ⋀∞ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of Δn, we obtain a decomposition of the basic representation L(Λ0) which induces the Kashiwara–Miwa–Stern decomposition of ⋀∞ and a construction of the canonical basis of ⋀∞ defined by Leclerc and Thibon in terms of certain monomial basis elements in D(Δn).

On the Hochschild (co)homology of a monomial algebra given by a cyclic quiver and two zero-relations

ABSTRACTLet K be an algebraically closed field and Γ a cyclic quiver. Xu and Wang investigated the Hochschild (co)homology groups of KΓ∕I, where I is an ideal of KΓ generated by one path. In this paper, in the case that I is an ideal of KΓ generated by two paths, we give the module structure of the Hochschild (co)homology groups of KΓ∕I.

Representation dimension of generalized path algebras of cyclic quivers

ABSTRACTLet Λ be a non semisimple symmetric artin algebra over a commutative artinian ring R and let repdim(Λ) be its representation dimension. Assume also that 𝒬 is a finite quiver whose connected components are amalgamations of single cycles possibly with multiple arrows, and that ℐ is an admissible ideal in the generalized path algebra Λ𝒬. We prove that repdim(Λ)+1.

Quantum groups via cyclic quiver varieties I

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig’s bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland’s recent work for type $\mathbb{A}\mathbb{D}\mathbb{E}$.