Pavel Etingof, Frederic Latour, Eric Rains
We determine the structure of the center and the trace space of the centrally extended preprojective algebra of an ADE quiver, introduced by the first and the third authors in math/0503393. It turns out that this structure has a mysterious relationship to the structure of the maximal nilpotent subalgebra of the corresponding simple Lie algebra.
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In this paper, we show that for a Koszul $n$-homogeneous algebra $Λ$, the quadratic dual of certain twisted trivial extension is the $(n+1)$-preprojective algebra of its quadratic dual, that is, $ (Δ_νΛ)^{!,op} \simeqΠ( Λ^{ !, op })$. This is applied to the $τ$-slice algebras of stable $n$-translation algebras and gives a noncommutative version of Bernstein-Gelfand-Gelfand correspondence for such algebras.
The Hilbert series of the preprojective algebras
本文利用带 Adams 分次的 dg 代数 Hilbert 级数公式，证明任意特征下预投射代数的 Hilbert 级数，并进一步计算其特殊中心扩张——箭图 Heisenberg 代数的 Hilbert 级数。
Morphisms and extensions between bricks over preprojective algebras of type A
本文利用“弧”计算A型预投射代数上brick之间的Hom空间与第一扩张空间的基，据此分类“弱例外序列”，并阐明其与A型遗传代数例外序列组合模型的关联。
In this article we study higher preprojective algebras, showing that various known results for ordinary preprojective algebras generalize to the higher setting. We first show that the quiver of the higher preprojective algebra is obtained by adding arrows to the quiver of the original algebra, and these arrows can be read off from the last term of the bimodule resolution of the original algebra. In the Koszul case we are able to obtain the new relations of the higher preprojective algebra by differentiating a superpotential and we show that when our original algebra is $d$-hereditary all the r…
Let A be the central extension of the preprojective algebra of an ADE quiver introduced by P. Etingof and E. Rains in math/0503393. The paper math/0606403 computes the structure of the zeroth Hochschild (co)homology of A. We generalize the results of math/0606403 by calculating the additive structure of all the Hochschild homology and cohomology groups of A and the cyclic homology of A, and to describe the universal deformation of A. Namely, we show that the (co)homology is periodic with period 4, and compute the first four (co)homology groups in each case.
We introduce a central extension of the preprojective algebra of a finite Dynkin quiver (depending on a regular weight for the corresponding root system), whose natural deformed version is flat (unlike that for the preprojective algebra). We calculate the Hilbert polynomial of the central extension, and show that it is a Frobenius algebra. As a corollary, we obtain the Hilbert series of the usual deformed preprojective algebra in which the deformation parameters are variables, and show that this algebra is Gorenstein (although it is not a flat module over the ring of parameters). The proofs ar…
The goal of this note two-fold. First, I draw attention to the way that semistability (in the sense of King) gives us a geometrical picture of (some of) the extension-closed abelian subcategories of a finite-dimensional algebra. Second, I describe Nathan Reading's shards of a hyperplane arrangement, and explain their relevance to understanding the semistability picture for finite-type preprojective algebras.
$2$-representation infinite algebras from non-abelian subgroups of $\operatorname{SL}_3$. Part I: Extensions of abelian groups
本文研究了从SL_3的非阿贝尔子群出发的3-前投射结构，证明R∗G具有此类结构当且仅当9整除群阶，并提供了新的2-表示无限代数类。
Determinantal modules over preprojective algebras and representations of Dynkin quivers
该研究利用箭图的Auslander-Reiten平移研究预投射代数上行列式模的扩张群，给出Dynkin情形下两量子子式乘积属对偶典范基的等价组合条件，并可检验相关量子丛单项式的拟交换性。
We give an interpretation of the $(q,t)$-deformed Cartan matrices of finite type and their inverses in terms of bigraded modules over the generalized preprojective algebras of Langlands dual type in the sense of Geiß-Leclerc-Schröer [Invent. math. 209 (2017)]. As an application, we compute the first extension groups between the generic kernels introduced by Hernandez-Leclerc [J. Eur. Math. Soc. 18 (2016)], and propose a conjecture that their dimensions coincide with the pole orders of the normalized $R$-matrices between the corresponding Kirillov-Reshetikhin modules.
We show that if H is a hereditary finite dimensional algebra, M is a finitely generated H-module and B is a semisimple subalgebra of the endomorphism algebra of M, then the representation dimension of the corresponding triangular matrix algebra is less or equal to 3 whenever one of the following conditions hold: i) H is of finite representation type; ii) H is tame and M is a direct sum of regular and preprojective modules; iii) M has no self-extensions
Quiver Heisenberg algebras: a cubic analogue of preprojective algebras
本文研究有向图海森堡代数作为预项目代数的中心扩张，探讨其在正特征下的表示理论，揭示其与路径代数的表示理论关系，并提供精确的Auslander-Reiten序列描述。
Lattice structure of Weyl groups via representation theory of preprojective algebras
本文利用预投射代数Π的表示论，研究有限Weyl群W上弱序的格同余的组合结构，建立了join-不可约同余、join-不可约（或meet-不可约）元素、不可分解τ-刚性（或τ⁻-刚性）模与Π的层之间的自然双射，并证明了Hasse箭图的两种标号方式一致。
We introduce and study a fermionization procedure for the cohomological Hall algebra $\mathcal{H}_{Π_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson--Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel-Moore homology of the stack of representations of the $μ$-deformed preprojective algebra introduced by Crawley-…
$2$-representation infinite algebras from non-abelian subgroups of $\operatorname{SL}_3$. Part II: Central extensions and exceptionals
本文研究了SL_3(C)中非阿贝尔子群作用下的斜群代数，探讨其作为3-前投射代数的条件，并通过计算麦基奎ver和cut，提供了新的2-表示无限代数例子。
For $(Q,W)$ a symmetric quiver with potential satisfying a Künneth-type condition, we construct (positive and negative) extensions of its K-theoretic Hall algebra which are bialgebras. In particular, there are bialgebra extensions of preprojective KHAs of a quiver $Q$ and one can construct their Drinfeld double algebra.
Let Q be a Dynkin quiver, and let P(Q) be the corresponding preprojective algebra. Let I be a set of pairwise different indecomposable irreducible components of varieties of P(Q)-modules such that generically there are no extensions between these components. We show that the number of elements in I is at most the number of positive roots of Q. Furthermore, we give a module theoretic interpretation of Leclerc's counterexample to a conjecture of Berenstein and Zelevinsky.
Rank $1$ modules are the building blocks of the category ${\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of a preprojective algebra of affine type $A$. Jensen, King and Su showed in \cite{JKS16} that the category ${\rm CM}(B_{k,n})$ provides an additive categorification of the cluster algebra structure on the coordinate ring $\mathbb C[{\rm Gr}(k, n)]$ of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb C^n$. Rank $1$ modules are indecomposable, they are known to be in bijection with $k$-subsets of $\{1,2,\dots,n\}$, and their explicit construction has…
$τ$-tilting theory and silting theory of skew group algebra extensions
本文研究斜群代数扩张的$\tau$-倾斜理论与沉默理论，建立了稳定支撑$\tau$-倾斜模与沉默复形偏序集之间的同构，并给出有限性与离散性的继承条件，应用于预投射代数等。
The goal of this paper is to explicitly describe in terms of generators and relations the universal central extension of the infinite dimensional Lie algebra, $\mathfrak{g} \otimes \mathbb{C}[t,t^{-1},u]$ with finite dimensional simple Lie algebra $\mathfrak{g}$ and $u^m=p(t)$.
We obtain a recurrent and monotone method for constructing and classifying nilpotent Lie algebras by means of successive central extensions. It consists in calculating the second cohomology of an extendable nilpotent Lie algebra with the subsequent study of the orbit space geometry of the automorphism group action on Grassmannians defined in terms of the second cohomology of the extendable nilpotent Lie algebra. Such a geometric method allows to classify some classes of nilpotent Lie algebras. The concept of a rigid central extension is introduced.
The purpose of this paper is to explicitly describe in terms of generators and relations the universal central extension of the infinite dimensional Lie algebra, $\mathfrak g\otimes \mathbb C[t,t^{-1},u|u^2=(t^2-b^2)(t^2-c^2)]$, appearing in the work of Date, Jimbo, Kashiwara and Miwa in their study of integrable systems arising from Landau-Lifshitz differential equation.
Three kinds of universal central extension are considered for a perfect Lie algebra. More precisely, one can consider such a Lie algebra as a Lie triple system, or a Leibniz algebra and construct appropriate central extensions. We show that Lie triple system universal central extension is isomorphic either to Lie or Leibniz algebra universal central extension considered as a Lie triple system in appropriate way.
In this paper we study universal central extensions and non-abelian tensor product of hom-Lie-Rinehart algebras. We discuss about universal $α$- central extensions, and, lifting of automorphisms and $α$-derivations to central extensions for hom-Lie-Rinehart algebras. This is in turn provide such lifting of automorphisms and $α^k$-derivations to the central extensions for hom-Lie algebras.
In this paper, we describe restricted one-dimensional central extensions of all finite dimensional simple restricted Lie algebras defined over fields of characteristic $p\ge 5$.
We describe how the center of the universal central extension of the genus zero Krichever-Novikov current algebra decomposes as a direct sum of irreducible modules for automorphism group of the coordinate ring of this algebra.
We generalize the linear algebra setting of Tate's central extension to arbitrary dimension. In general, one obtains a Lie (n+1)-cocycle. We compute it explicitly. The construction is based on a Lie algebra variant of Beilinson's adelic multidimensional residue symbol, generalizing Tate's approach to the local residue symbol for 1-forms on curves.
Let $p(t)\in\mathbb C[t]$ be a polynomial with distinct roots and nonzero constant term. We describe, using Faá de Bruno's formula and Bell polynomials, the universal central extension in terms of generators and relations for the hyperelliptic current Lie algebras $\mathfrak g\otimes R$ whose coordinate ring is of the form $R=\mathbb C[t,t^{-1},u\,|\, u^2=p(t)]$.
Dialgebras are generalizations of associative algebras which give rise to Leibniz algebras instead of Lie algebras. In this paper we study super dialgebras and Leibniz superalgebras, which are $\z_2$-graded dialgebras and Leibniz algebras. We also study universal central extesions of Leibniz superalgebras and obtain some results as in the case of Lie superalgebras. We determine universal central extensions of basic classical Lie superalgebras in the category of Leibniz superalgebras. They play a key role in studying Leibniz superalgebras graded by finite root systems.
On some results of Harish-Chandra for representations of p-adic groups, extended to their central extensions
本文完成哈里-查兰德拉关于p-进群的parabolic诱导不可约性与mu函数解析行为的联系证明，并展示该证明在中心扩张情况下的有效性。
Consider the maximal nilpotent subalgebra $n_+(A_1^{(1)})$ of the simplest affine algebra $A_1^{(1)}$ which is one of the $\mathbb{N}$-graded Lie algebras with minimal number of generators. We show truncated versions of this algebra in positive characteristic admit the structure of a family of restricted Lie algebras. We compute the ordinary and restricted 1- and 2-cohomology spaces with trivial coefficients by giving bases. With these we explicitly describe the restricted 1-dimensional central extensions.
In these notes, we give a survey of the main results of [BC] and [BW]. Our aim is to generalize the geometric classification of (one-sided) ideals of the first Weyl algebra $ A_1(C) $ (see [BW1, BW2]) to the ring $ D(X) $ of differential operators on an arbitrary complex smooth affine curve X. We approach this problem in two steps: first, we classify the ideals of D(X) up to stable isomorphism, in terms of the Picard group of X; then, we refine this classification by describing each stable isomorphism class as a disjoint union of (certain quotients of) generalized Calogero-Moser spaces C_n(X, …
A note on the representation theory of central extensions of reductive $p$-adic groups
本文验证了约化p进群表示理论中的若干基本结果可推广到其有限中心扩张，为相关表示论研究提供了更一般的适用框架。
Motivated by positive energy representations, we classify those continuous central extensions of the compactly supported gauge Lie algebra that are covariant under a 1-parameter group of transformations of the base manifold.
The universal central extensions and their extension kernels of the matrix Lie superalgebra sl(m, n, A), the Steinberg Lie superalgebra st(m, n, A) in category {\bf SLeib} of Leibniz superalgebras are determined under a weak assumption (compared with \cite{MP}) using the first Hochschild homology and the first cyclic homology group.
We give a proof of the fact that a simply-connected symplectic homogeneous space $(M,ω)$ of a connected Lie group $G$ is the universal cover of a coadjoint orbit of a one-dimensional central extension of $G$. We emphasise the rôle of symplectic group cocycles and the relationship between such cocycles, left-invariant presymplectic structures on $G$ and central extensions of $G$; in particular, we show that integrability of a central extension of $\mathfrak{g}$ to a central extension of $G$ is equivalent to integrability of a representative Chevalley-Eilenberg 2-cocycle of $\mathfrak{g}$ to a s…
In this paper, using a quantum superalgebra associated with the universal central extension of sl(2,2)^{(1)}, we introduce new R-matrices having an extra parameter x. As x\to 0, they become those associated with the symmetric and anti-symmetric tensor products of the copies of the vector representation of U_q sl(2,2)^{(1)}.
We consider central extensions $Z\hookrightarrow E\twoheadrightarrow G$ in the category of linear differential algebraic groups. We show that if $G$ is simple non-commutative and $Z$ is unipotent with the differential type smaller than that of $G$, then such an extension splits. We also give a construction of central extensions illustrating that the condition on differential types is important for splitting. Our results imply that non-commutative almost simple linear differential algebraic groups, introduced by Cassidy and Singer, are simple.
We study canonical central extensions of the general linear group of the ring of adeles on a smooth projective algebraic surface $X$ by means of the group of integers. By these central extensions and adelic transition matrices of a rank $n$ locally free sheaf of ${\mathcal O}_X$-modules we obtain the local (adelic) decomposition for the difference of Euler characteristics of this sheaf and the sheaf ${\mathcal O}_X^n$. Two various calculations of this difference lead to the Riemann-Roch theorem on $X$ (without the Noether formula).
Let $Γ$ denote a central extension of the form $1\to \mathbb{Z}^r\toΓ\to \mathbb{Z}^n\to 1$. In this paper we describe the topology of the spaces of homomorphisms $\text{Hom}(Γ, U(m))$ and the associated moduli spaces $\text{Rep}(Γ, U(m))$, where $U(m)$ is the group of $m\times m$ unitary matrices.
Let $\mathbb{G}$ be a Lie group with solvable connected component and finitely-generated component group and $α\in H^2(\mathbb{G},\mathbb{S}^1)$ a cohomology class. We prove that if $(\mathbb{G},α)$ is of type I then the same holds for the finite central extensions of $\mathbb{G}$. In particular, finite central extensions of type-I connected solvable Lie groups are again of type I. This is by contrast with the general case, whereby the type-I property does not survive under finite central extensions. We also show that ad-algebraic hulls of connected solvable Lie groups operate on these even wh…
These are the notes for some lectures given by this author at Harish-Chandra Research Institute, Allahabad in March 2014 for a workshop on Schur multipliers. The lectures aimed at giving an overview of the subject with emphasis on groups of Lie type over finite, real and $p$-adic fields.
We introduce a class of novel $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded color superalgebras of infinite dimension. It is done by realizing each member of the class in the universal enveloping algebra of a Lie superalgebra which is a module extension of the Virasoro algebra. Then the complete classification of central extensions of the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded color superalgebras is presented. It turns out that infinitely many members of the class have non-trivial extensions. We also demonstrate that the color superalgebras (with and without central extensions) have adjoint and…
Given any representation of an arbitrary Lie algebra L over a field k of characteristic 0, we construct representations of L on bosonic and fermionic Fock space. The method gives an explicit formula for a (sometimes trivial) 2-cocycle in H^2(L;k). We illustrate these techniques with several concrete examples.
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