Christof Geiss, Bernard Leclerc, Jan Schröer
We realize the enveloping algebra of the positive part of a symmetrizable Kac-Moody algebra as a convolution algebra of constructible functions on module varieties of some Iwanaga-Gorenstein algebras of dimension 1.
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We generalize Lusztig's nilpotent varieties, and Kashiwara and Saito's geometric construction of crystal graphs from the symmetric to the symmetrizable case. We also construct semicanonical functions in the convolution algebras of generalized preprojective algebras. Conjecturally these functions yield semicanonical bases of the enveloping algebras of the positive part of symmetrizable Kac-Moody algebras.
To each symmetrizable Cartan matrix, we associate a finite free EI category. We prove that the corresponding category algebra is isomorphic to the algebra defined in [C. Geiss, B. Leclerc, and J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), 61--158], which is associated to another symmetrizable Cartan matrix. In certain cases, the algebra isomorphism provides an algebraic enrichment of the well-known correspondence between symmetrizable Cartan matrices and graphs with automorphisms.
We generalize the Caldero-Chapoton formula for cluster algebras of finite type to the skew-symmetrizable case. This is done by replacing representation categories of Dynkin quivers by categories of locally free modules over certain Iwanaga-Gorenstein algebras introduced in Part I. The proof relies on the realization of the positive part of the enveloping algebra of a simple Lie algebra of the same finite type as a convolution algebra of constructible functions on representation varieties of $H$, given in Part III. Along the way, we obtain a new result on the PBW basis of this convolution algeb…
We give an overview of our effort to introduce (dual) semicanonical bases in the setting of symmetrizable Cartan matrices.
The Aizenbud-Lapid binary operation for symmetrizable Cartan types
本文将Aizenbud和Lapid近期为对称Cartan矩阵关联晶体图$B(-\infty)$引入的二元运算，推广至可对称化Cartan矩阵。
Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac-Moody algebra. The underlying set consists of the irreducible components of Lusztig's quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac-Moody algebras by replacing Lusztig's preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.
We introduce and study a class of Iwanaga-Gorenstein algebras defined via quivers with relations associated with symmetrizable Cartan matrices. These algebras generalize the path algebras of quivers associated with symmetric Cartan matrices. We also define a corresponding class of generalized preprojective algebras. Without any assumption on the ground field, we obtain new representation-theoretic realizations of all finite root systems.
We realize the enveloping algebra of the positive part of a semisimple complex Lie algebra as a convolution algebra of constructible functions on module varieties of some Iwanaga-Gorenstein algebras of dimension 1.
For any given symmetrizable Cartan matrix $C$ with a symmetrizer $D$, Geiß~ et al. (2016) introduced a generalized preprojective algebra $Π(C, D)$. We study tilting modules and support $τ$-tilting modules for the generalized preprojective algebra $Π(C, D)$ and show that there is a bijection between the set of all cofinite tilting ideals of $Π(C,D)$ and the corresponding Weyl group $W(C)$ provided that $C$ has no component of Dynkin type. When $C$ is of Dynkin type, we also establish a bijection between the set of all basic support $τ$-tilting $Π(C,D)$-modules and the corresponding Weyl group $…
For $k \ge 1$ we consider the $K$-algebra $H(k) := H(C,kD,Ω)$ associated to a symmetrizable Cartan matrix $C$, a symmetrizer $D$, and an orientation $Ω$ of $C$, which was defined in Part 1. We construct and analyse a reduction functor from rep$(H(k))$ to rep$(H(k-1))$. As a consequence we show that the canonical decomposition of rank vectors for $H(k)$ does not depend on $k$, and that the rigid locally free $H(k)$-modules are up to isomorphism in bijection with the rigid locally free $H(k-1)$-modules. Finally, we show that for a rigid locally free $H(k)$-module of a given rank vector the Euler…
Skew-symmetrizable cluster algebras from surfaces and symmetric quivers
本文研究了从表面和对称四元组中得到的非对称簇代数，通过几何方法和表示论建立了与正交不可分解模的对应关系，并提供了Caldero-Chapoton映射。
Lusztig sheaves and integrable highest weight modules in the symmetrizable case
本文基于前人工作，研究对称可约情况下的Lusztig sheaves与可积分最高权模的构造及格罗滕迪克群的实现，揭示其与晶格结构的关系。
In this paper, we give a description of the skew-symmetrizable matrices and their mutation classes which are determined by the generalized Cartan matrices of affine type.
We introduce a family of quiver Hecke algebras which give a categorification of quantum Borcherds algebra associated to an arbitrary Borcherds-Cartan datum.
The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a c…
Let U be the quantum group associated to a symmetrizable generalized Cartan matrix. We give a realization of U from the category of the representations of certain product valued quiver.
$q$-Coxeter matrix and $q$-Cartan matrix for a homogeneous bound quiver
本文为齐次有界箭图引入 $q$-Coxeter 矩阵，并建立其与 $q$-Cartan 矩阵的关系。
Motivated by the mutation theory of quivers with potentials developed by Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster algebras it provides, we propose a mutation theory of species with potentials for species that arise from skew-symmetrizable matrices that admit a skew-symmetrizer with pairwise coprime diagonal entries. The class of skew-symmetrizable matrices covered by the mutation theory proposed here contains a class of matrices that do not admit global unfoldings, that is, unfoldings compatible with all possible sequences of mutations.
Skew group categories, algebras associated to Cartan matrices and folding of root lattices
本文研究了有限群作用下斜群范畴与Cartan矩阵代数的Morita等价关系，并通过范畴化方法实现了根格子的折叠投影。
Henriques and Kamnitzer have defined a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra that gives it the structure of a coboundary category (somewhat analogous to a braided monoidal category). Kamnitzer and Tingley then gave an alternative definition of the crystal commutor, using Kashiwara's involution on Verma crystals, that generalizes to the setting of symmetrizable Kac-Moody algebras. In the current paper, we give a geometric interpretation of the crystal commutor using quiver varieties. Equipped with this interpretation we show that the commuto…
Quiver Hecke algebras for Borcherds-Cartan datum II
本文给出了箭图Hecke代数不可约模Grothendieck群的晶体结构，实现了量子Borcherds代数晶体$B(\infty)$与$B(\lambda)$的范畴化，并研究了不可约最高权模的循环范畴化。
Mutation Of Matrices Over Group Rings
本文研究了群环上斜对称可化矩阵的突变定义及其与具有对称性的有向无环图的突变关系，并提出了新的突变规则。
We study the class of weighted locally gentle quivers. This naturally extends the class of gentle quivers and gentle algebras, which have been intensively studied in the representation theory of finite-dimensional algebras, to a wider class of potentially infinite-dimensional algebras. Weights on the arrows of these quivers lead to gradings on the corresponding algebras. For the natural grading by path lengths, any locally gentle algebra is a Koszul algebra. Our main result is a general combinatorial formula for the determinant of the weighted Cartan matrix of a weighted locally gentle quiver.…
Reflection functors on quiver Hecke algebras
本文针对任意可对称化 Kac-Moody 型的箭图 Hecke 代数构造反射函子，并证明其范畴化了 Lusztig 的辫子对称性。
We completely describe presentations of Lie superalgebras with Cartan matrix if they are simple Z-graded of polynomial growth. Such matrices can be neither integer nor symmetrizable. There are non-Serre relations encountered. In certain cases there are infinitely many relations. Our results are applicable to the Lie algebras with the same Cartan matrices as the Lie superalgebras considered.
Mutation of signed valued quivers and presentations of simple complex Lie algebras
本文提出带符号估值有向图的突变规则，并通过构造李代数的方式呈现了简单复李代数的结构。
That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of the most elegant results in mathematics. The definition of the root system is done by fixing a Cartan subalgebra of the given Lie algebra. The remarkable fact is that (up to isomorphism) this construction is independent of the choice of the Cartan subalgebra. The modern way of establishing this fact is by showing that all Cartan subalgebras are conjugate. For …
Let $\fg$ be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra $\fh$ and the Weyl group $W$. Let $P_+$ be the set of dominant integral weights. For $λ\in P_+$, let $L(λ)$ be the irreducible, integrable, highest weight representation of $\fg$ with highest weight $λ$. For a positive integer $s$, define the {\em saturated tensor semigroup} as \begin{align*} Γ_s:= \{(λ_1, \dots, λ_s,μ)\in P_+^{s+1}: \exists\, N>1 \,\,\text{with}\,\, L(Nμ)\subset L(Nλ_1)\otimes \dots \otimes L(Nλ_s)\}. \end{align*} The aim of this paper is to begin a systematic study of $Γ_s$ in the infinite…
Generic bases of skew-symmetrizable affine type cluster algebras
本文研究了仿射类型簇代数的基，通过Caldero-Chapoton公式与簇变量的Laurent表达式一致，得到包含所有簇单项式的基。
$t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras
本文证明任意有限型的 $t$-量子化 Cartan 矩阵可决定 quiver Hecke 代数 cuspidal 模间 R-矩阵的不变量，并利用 Dynkin quiver 的组合 AR-quiver 作为证明关键。
A cluster algebra is a commutative algebra whose structure is decided by a skew-symmetrizable matrix or a quiver. When a skew-symmetrizable matrix is invariant under an action of a finite group and this action is admissible, the folded cluster algebra is obtained from the original one. Any cluster algebra of non-simply-laced affine type can be obtained by folding a cluster algebra of simply-laced affine type with a specific $G$-action. In this paper, we study the combinatorial properties of quivers in the cluster algebra of affine type. We prove that for any quiver of simply-laced affine type,…
A new class of algebras have been introduced by Khovanov and Lauda and independently by Rouquier. These algebras categorify one-half of the Quantum group associated to arbitrary Cartan data. In this paper, we use the combinatorics of Lyndon words to construct the irreducible representations of those algebras associated to Cartan data of finite type. This completes the classification of simple modules for the quiver Hecke algebra initiated by Kleshchev and Ram.
The $(q,t)$-Cartan matrix specialized at $t=1$, usually called the quantum Cartan matrix, has deep connections with (i) the representation theory of its untwisted quantum affine algebra, and (ii) quantum unipotent coordinate algebra, root system and quantum cluster algebra of kew-symmetric type. In this paper, we study the $(q,t)$-Cartan matrix specialized at $q=1$, called the $t$-quantized Cartan matrix, and investigate the relations with (ii') its corresponding quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetrizable type.
Conjugation Theorem For Affine Kac-Moody Superalgebras
本文将Cartan子代数的共轭定理从对称可约Kac-Moody代数推广到仿射Kac-Moody超代数，并深入探讨了其结构及虚根空间的关系。
Varagnolo-Vasserot and Rouquier proved that, in a symmetric generalized Cartan matrix case, the simple modules over the quiver Hecke algebra with a special parameter correspond to the upper global basis. In this note we show that the simple modules over the quiver Hecke algebras with a generic parameter also correspond to the upper global basis in a symmetric generalized Cartan matrix case.
Inspired by Fock-Goncharov's amalgamation procedure \cite{Fock-Goncharov-2006}, Shen-Weng introduced string diagrams in \cite{Shen-Weng-2021}, which are very useful to describe many interesting skew-symmetrizable matrices closely related with Lie theory. In this paper, we prove that the skew-symmetrizable matrices from string diagrams are in the smallest class $\mathcal P^\prime$ of skew-symmetrizable matrices containing the $1\times 1$ zero matrix and closed under mutations and source-sink extensions. This result applies to the exchange matrices of cluster algebras from double Bruhat cells, u…
In this paper, we prove Khovanov-Lauda's cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras. Let $U_q(g)$ be the quantum group associated with a symmetrizable Cartan datum and let $V(Λ)$ be the irreducible highest weight $U_q(g)$-module with a dominant integral highest weight $Λ$. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra $R^Λ$ gives a categorification of $V(Λ)$.
To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the integers attached on the vertices are all equal to one. The construction of reflection functors for quiver varieties are generalized to our case, in which these relate to simple reflections in the Weyl group of some symmetrizable, possibly non-symmetric Kac-Moody algebra. The moduli spaces of meromorphic connections on the rank 2 trivial bundle over the Riemann…
Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay-Cartan matrices for finite subgroups G of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights. In the McKay-Cartan case for finite subgroups of the special linear group, the cokernel is related to the abelianization of the subgroup G. In the special case of the classica…
This paper aims at a geometric realization of the Yangian of non-simply laced type in terms of quiver with potentials. For every quiver with symmetrizer, there is an extended quiver with superpotential, whose Jacobian algebra is the generalized preprojective algebra of Geiß, Leclerc, and Schröer arXiv:1410.1403. We study the cohomological Hall algebra of Kontsevich and Soibelman associated to this quiver with potential. In particular, we prove a dimensional reduction result, and provide a shuffle formula of this cohomological Hall algebra. In the case when the quiver with symmetrizer comes fro…
Construction of PBW and canonical bases in the folding extension algebras for symmetrizable types
研究源于Lusztig构造的扩张代数，扩展Varagnolo和Vasserot方法构造斜群代数，证明相关过渡矩阵性质，为对称型情形下PBW基与典范基及其关系提供几何范畴化。
The preprojective algebra of a finite EI quiver
本文定义了有限EI箭图的预投射代数，证明其同构于某个张量代数，并进一步证明Cartan型有限EI箭图对应的预投射代数同构于广义预投射代数。
This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. As a consequence, for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the gr…
Deformed Cartan matrices and generalized preprojective algebras II: General type
提出带多参数的变形对称化广义Cartan矩阵定义，借助广义预投射代数分次模给出范畴解释，推导逆矩阵的辫群组合公式，并在特定条件下与质量变形Cartan矩阵一致。
Tame quivers and affine bases II: nonsimply-laced cases
本文用Ringel-Hall代数方法将仿射典范基推广到一般对称化情形，证明所得bar不变基与Lusztig带符号典范基一致，并在$\tilde{B}_n,\tilde{C}_n$型中直接给出典范基。
Quiver Bases of Cartan Squares of Minuscule Representations
本文为极小表示的Cartan平方构造箭图基，统一证明任意极小偏序集上分段线性Coxeter运动与行运动满足循环筛选现象，并指出典范基仅与A型长循环相容。
We study the cocenter of the cyclotomic quiver Hecke algebra $R^Λ_α$ associated to an {\it arbitrary} symmetrizable Cartan matrix $A=(a_{ij})_{i,j}\in I$, $Λ\in P^+$ and $α\in Q_n^+$. We introduce a notion called "piecewise dominant sequence" and use it to construct some explicit homogeneous elements which span the maximal degree component of the cocenter of $R^Λ_α$. We show that the minimal degree components of the cocenter of $R^Λ_α$ is spanned by the image of some KLR idempotent $e(ν)$, where each $ν\in I^α$ is piecewise dominant. As an application, we show that the weight space $L(Λ)_{Λ-α}…
This paper generalizes former works of Derksen, Weyman and Zelevinsky about quivers with potentials. We consider semisimple finite-dimensional algebras $E$ over a field $F$, such that $E \otimes_{F} E^{op}$ is semisimple. We assume that $E$ contains a certain type of $F$-basis which is a generalization of a multiplicative basis. We study potentials belonging to the algebra of formal power series, with coefficients in the tensor algebra over $E$, of any finite-dimensional $E$-$E$-bimodule on which $F$ acts centrally. In this case, we introduce a cyclic derivative and to each potential we associ…
We introduce the notions of quasi-Laurent and Laurent families of simple modules over quiver Hecke algebras of arbitrary symmetrizable types. We prove that such a family plays a similar role of a cluster in the quantum cluster algebra theory and exhibits a quantum Laurent positivity phenomenon for the basis of the quantum unipotent coordinate ring $\mathcal{A}_q(\mathfrak{n}(w))$, coming from the categorification. Then we show that the families of simple modules categorifying GLS-clusters are Laurent families by using the PBW-decomposition vector of a simple module $X$ and categorical interpre…
We introduce and study potentials, mutations and Jacobian algebras in the framework of tensor algebras associated with symmetrizable dualizing pairs of bimodules on a symmetric algebra over any commutative ground ring. The graded context is also considered by starting from graded bimodules, and the classical non simply-laced context of modulated quivers with potentials is a particular case. The study of potentials in this framework is related to symmetrically separable algebras, and we have two kinds of potentials: the symmetric and the non symmetric ones. When the Casimir ideal of the symmetr…
A contragredient Lie superalgebra is a superalgebra defined by a Cartan matrix. A contragredient Lie superalgebra has finite-growth if the dimensions of the graded components (in the natural grading) depend polynomially on the degree. In this paper we classify finite-growth contragredient Lie superalgebras. Previously, such a classification was known only for the symmetrizable case.
We consider a generalization of the quiver varieties of Lusztig and Nakajima to the case of all symmetrizable Kac-Moody Lie algebras. To deal with the non-simply laced case one considers admissible automorphisms of a quiver and the irreducible components of the quiver varieties fixed by this automorphism. We define a crystal structure on these irreducible components and show that the crystals obtained are isomorphic to those associated to the crystal bases of the lower half of the universal enveloping algebra and the irreducible highest weight representations of the non-simply laced Kac-Moody …
Extensions of Restricted Simple Modules for Restricted Lie Algebras of Cartan Type
本文确定了特征 p≥5 的 Cartan 型受限单李代数受限单模的一阶扩张群，回答了 Benkart-Feldvoss 问题，并通过有限受限上同调复形计算扩张，给出了例外模的闭式公式，且确定了特征七下 S(3;1)^(1) 的箭图。
Let $\mathfrak{g}$ be an affine Kac-Moody algebra with symmetric Cartan datum, $\mathfrak{n^{+}}$ be the maximal nilpotent subalgebra of $\mathfrak{g}$. By the Hall algebra approach, we construct integral bases of the $\mathbb{Z}$-form of the enveloping algebra $U(\mathfrak{n^{+}})$. In particular, the representation theory of tame quivers is essentially used in this paper.
A Littelmann path model is constructed for crystals pertaining to a not necessarily symmetrizable Borcherds-Cartan matrix. Here one must overcome several combinatorial problems coming from the imaginary simple roots. The main results are an isomorphism theorem and a character formula of Borcherds-Kac-Weyl type for the crystals. In the symmetrizable case, the isomorphism theorem implies that the crystals constructed by this path model coincide with those of Jeong, Kang, Kashiwara and Shin obtained by taking the limit at q=0 in the quantized enveloping algebra.
Expander representations of quivers
提出箭图的扩张表示定义以细化斜率稳定性，通过一般子表示概念和嘉当矩阵谱性质，证明代数闭域上任意野箭图存在一致扩张表示。
The first goal of this note is to extend the well-known Feigin homomorphisms taking quantum groups to quantum polynomial algebras. More precisely, we define generalized Feigin homomorphisms from a quantum shuffle algebra to quantum polynomial algebras which extend the classical Feigin homomorphisms along the embedding of the quantum group into said quantum shuffle algebra. In a recent work of Berenstein and the author, analogous extensions of Feigin homomorphisms from the dual Hall-Ringel algebra of a valued quiver to quantum polynomial algebras were defined. To relate these constructions, we …
Cartan matrices are of fundamental importance in representation theory. For algebras defined by quivers (i.e. directed graphs) with relations the computation of the entries of the Cartan matrix amounts to counting nonzero paths in the quivers, leading naturally to a combinatorial setting. In this paper we study a refined version, so-called q-Cartan matrices, where each nonzero path is weighted by a power of an indeterminant q according to its length. Specializing q=1 gives the classical Cartan matrix. Our main motivation are derived module categories and their invariants: the invariant factors…
Recently it has been noticed that many interesting combinatorial objects belong to a class of semigroups called left regular bands, and that random walks on these semigroups encode several well-known random walks. For example, the set of faces of a hyperplane arrangement is endowed with a left regular band structure. This paper studies the module structure of the semigroup algebra of an arbitrary left regular band, extending results for the semigroup algebra of the faces of a hyperplane arrangement. In particular, a description of the quiver of the semigroup algebra is given and the Cartan inv…
Stabilization phenomena in Kac-Moody algebras and quiver varieties
本文研究可对称化 Kac-Moody 代数中，通过延伸子图 X_0 扩展 Dynkin 图时，分支重数与张量积重数的稳定化现象，并证明权重数关于 X_0 长度呈多项式，推广了 Kleber-Viswanath 与 Benkart 等人的结果。
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