Dror Bar-Natan, Roland van der Veen
We find and discuss an unexpected (to us) order $n$ cyclic group of automorphisms of the Lie algebra $I\mathfrak{u}_n := \mathfrak{u}_n\ltimes\mathfrak{u}_n^\ast$, where $\mathfrak{u}_n$ is the Lie algebra of upper triangular $n\times n$ matrices. Our results also extend to $gl_{n+}^ε$, a ``solvable approximation'' of $gl_n$, as defined within.
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In this paper we classify all the cyclic finite dimensional indecomposable\\ modules of the perfect Lie algebras $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$, given by the semidirect sum of the simple Lie algebra $A_n$ with its standard representation. Furthermore, using the embeddings of the Lie algebras $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$ in $\mathfrak{sl}(n+2)$, we show that any finite dimensional irreducible module of $\mathfrak{sl}(n+2)$ restricted to $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$ is a cyclic module and that any cyclic $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1…
Quantum $α$-determinant cyclic modules of $\mathcal{U}_q(\mathfrak{gl}_n)$
作为量子行列式的一种特殊的单参数变形，我们引入了量子$α$ -行列式，并研究了由它生成的$\mathcal{U}_q(\mathfrak{gl}_n)$ -循环模：证明了该循环模中每个不可约表示的重数由一个称为$q$ -内容判别式的多项式决定.本结果的一部分是Matsumoto和Wakayama（2005）结果的量子对应，然而，在我们的情况下出现了一个新的显著特征。
Cyclic wide subalgebras of semisimple Lie algebras
本文研究半单李代数的循环宽子代数，给出正则Levi可分解子代数为循环宽的判定条件，并在根基为ad幂零子代数的情形下证明该条件是充要的。
Let $A$ be the path algebra of a Dynkin quiver $Q$ over a finite field, and $\mathscr{P}$ be the category of projective $A$-modules. Denote by $C^1(\mathscr{P})$ the category of 1-cyclic complexes over $\mathscr{P}$, and $\tilde{\mathfrak{n}}^+$ the vector space spanned by the isomorphism classes of indecomposable and non-acyclic objects in $C^1(\mathscr{P})$. In this paper, we prove the existence of Hall polynomials in $C^1(\mathscr{P})$, and then establish a relationship between the Hall numbers for indecomposable objects therein and those for $A$-modules. Using Hall polynomials evaluated at…
Given a complex semisimple Lie algebra ${\mathfrak g}$ and a commutative ${\mathbb C}$-algebra $A$, let ${\mathfrak g}[A] = {\mathfrak g} \otimes A$ be the corresponding generalized current algebra. In this paper we explore questions involving the computation and finite-dimensionality of extension groups for finite-dimensional ${\mathfrak g}[A]$-modules. Formulas for computing $\operatorname{Ext}^{1}$ and $\operatorname{Ext}^{2}$ between simple ${\mathfrak g}[A]$-modules are presented. As an application of these methods and of the use of the first cyclic homology, we completely describe $\oper…
Faithful Cyclic Modules for Enveloping Algebras and Sklyanin Algebras
设$U$是特征为零的域上有限维非交换李代数$\mathfrak{g}$的包络代数。证明了存在$U_1 = \mathfrak{g}\oplus K$的开非空开子集$X$，使得$U/Ux$对所有$x \in X$都是忠实的.我们证明了类似的结果均匀化包络代数和三维Sklyanin代数在无穷阶点。了解这些结果是否有一个共同的概括是很有意思的。
Let $\mathfrak g$ be a finite-dimensional Lie algebra. The symmetric algebra $\mathcal S(\mathfrak g)$ is equipped with the standard Lie-Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on $\mathcal S(\mathfrak g)$ to any finite order automorphism $θ$ of $\mathfrak g$. We study related Poisson-commutative subalgebras $\mathcal C$ of $\mathcal S(\mathfrak g)$ and associated Lie algebra contractions of $\mathfrak g$. To obtain substantial results, we have to assume that $\mathfrak g$ is semisimple. Then w…
We construct a large class of examples of the cyclic sieving phenomenon by expoiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the associated Kashiwara crystal. For $r\ge 0$, the triple $(X,c,P)$ which exhibits the cyclic sieving phenomenon is constructed as follows: the set $X$ is the set of isolated vertices in the crystal $\otimes^rB$; the map $c\colon X\rightarrow X$ is a generalisation of promotion acting on standard tableaux of rectangular shape and the polynomial $P$ is the fake degre…
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…
We find the normal form of nilpotent elements in semisimple Lie algebras that generalizes the Jordan normal form in $\mathfrak{sl}_N$, using the theory of cyclic elements.
For a classical group $G$ over a field $F$ together with a finite-order automorphism $θ$ that acts compatibly on $F$, we describe the fixed point subgroup of $θ$ on $G$ and the eigenspaces of $θ$ on the Lie algebra $\mathfrak{g}$ in terms of cyclic quivers with involution. More precise classification is given when $\mathfrak{g}$ is a loop Lie algebra, i.e., when $F=\mathbb{C}((t))$.
On Zeta functions and $μ$-series of string algebras
本文研究弦代数的ζ函数与μ级数，证明了弦代数的素数定理，并证明非本土弦代数具有指数增长，且弦代数本土当且仅当其μ级数有理。
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.
For the cyclic group $\mathbb{Z}_3$ and positive integer $k$, we study the representations of the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$. All the irreducible modules for $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are classified and constructed explicitly. Quantum dimensions and fusion rules for the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are completely determined.
Braid group actions, Baxter polynomials, and affine quantum groups
本文通过修正经典构造，将辫群作用提升到Yangian交换子代数，并通过对偶与量子环代数联系，证明了循环性条件的等价性。
A Hesselink-type formula for the nilpotent cone of Lie algebra representations
针对平凡型非循环扩展箭图表示，作者证明了对应幂零锥的q特征公式，还定义了Hesselink型表示并提出相关猜想、给出几何解释。
Let ${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ be the double Ringel--Hall algebra of the cyclic quiver $\triangle(n)$ and let $\dot{\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ be the modified quantum affine algebra of ${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$. We will construct an integral form $\dot{{\mathfrak D}_{\vartriangle}}(n)$ for $\dot{\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ such that the natural algebra homomorphism from $\dot{{\mathfrak D}_{\vartriangle}}(n)$ to the integral affine quantum Schur algebra is surjective. Furthermore, we will use Hall algebras to construct…
Let $\mathfrak{g}_{\mathbb{R}}$ be a split real, simple Lie algebra with complexification $\mathfrak{g}$. Let $G_{\mathbb{C}}$ be the connected, simply connected Lie group with Lie algebra $\mathfrak{g}$, $G_{\mathbb{R}}$ the connected subgroup of $G_{\mathbb{C}}$ with Lie algebra $\mathfrak{g}_{\mathbb{R}}$, and $G$ a covering group of $G_{\mathbb{R}}$ with a maximal compact subgroup $K$. A complete classification of "small" $K$ types is derived via Clifford algebras, and an analog, $P^ξ$, of Kostant's $P^γ$ matrix is defined for a $K$ type $ξ$ of principal series admitting a small $K$ type. …
A uniform construction of Chevalley normal forms for automorphic Lie algebras on the Riemann sphere
本研究针对SU(2)有限子群相关的自守李代数，统一构造了不变量模生成元与缠结算子，给出单李代数情形下类Chevalley基的生成元，解释了不同对称群自守李代数同构现象，并算出例外李型的结构常数。
If $\mathfrak{g} \subseteq \mathfrak{h}$ is an extension of Lie algebras over a field $k$ such that ${\rm dim}_k (\mathfrak{g}) = n$ and ${\rm dim}_k (\mathfrak{h}) = n + m$, then the Galois group ${\rm Gal} \, (\mathfrak{h}/\mathfrak{g})$ is explicitly described as a subgroup of the canonical semidirect product of groups ${\rm GL} (m, \, k) \rtimes {\rm M}_{n\times m} (k)$. An Artin type theorem for Lie algebras is proved: if a group $G$ whose order isinvertible in $k$ acts as automorphisms on a Lie algebra $\mathfrak{h}$, then $\mathfrak{h}$ is isomorphic to a skew crossed product $\mathfrak…
Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$ generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of pairwise distinct complex numbers $z_1,\ldots,z_N$. We prove that this subalgebra has a cyclic vector in the space of singular vectors of the tensor product of any finite-dimensional irreducible $\mathfrak{g}$-modules, for all values of the parameters $z_1,\ldots,z_N$. We deduce from this result the Bethe Ansatz conjecture in the Feigin-Frenkel form which states…
For a semisimple Lie algebra $\mathfrak{g}$ of rank $n$, let $\overline{U}_ζ(\mathfrak{g})$ be the restricted quantum group of $\mathfrak{g}$ at a primitive fourth root of unity. This quantum group admits a natural Borel-induced representation $V({\boldsymbol{t}})$, with ${\boldsymbol{t}}\in(\mathbb{C}^\times)^n$ determined by a character on the Cartan subalgebra. Ohtsuki showed that for $\mathfrak{g}=\mathfrak{sl}_2$, the braid group representation determined by tensor powers of $V({\boldsymbol{t}})$ is the exterior algebra of the Burau representation. In this paper, we generalize the tensor …
$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs II: injectivity
本文证明分裂型$\imath$量子环代数到加权射影线$\imath$Hall代数的态射在有限或仿射型下是单射，并利用循环箭图的$\imath$Hall代数实现仿射$\mathfrak{gl}_n$的$\imath$量子环代数。
Dualities of Gaudin models with irregular singularities for general linear Lie (super)algebras
本文证明了具有不规则奇点的Gaudin模型中$\mathfrak{gl}_d$与$\mathfrak{gl}_{p+m|q+n}$的双重性，并展示了其在无限维模空间上的作用特性。
We consider the Whittaker modules $M_{1}(λ,μ)$ for the Weyl vertex algebra $M$, constructed in arXiv:1811.04649, where it was proved that these modules are irreducible for each finite cyclic orbifold $M^{\Bbb Z_n}$. In this paper, we consider the modules $M_{1}(λ,μ)$ as modules for the ${\Bbb Z}$-orbifold of $M$, denoted by $M^0$. $M^0$ is isomorphic to the vertex algebra $\mathcal W_{1+\infty, c=-1} = \mathcal M(2) \otimes M_1(1)$ which is the tensor product of the Heisenberg vertex algebra $M_1(1)$ and the singlet algebra $\mathcal M(2)$. Furthermore, these modules are also modules of the Li…
On the supersymmetric XXX spin chains associated to $\mathfrak{gl}_{1|1}$
本文研究 gl(1|1) 超对称 XXX 自旋链，显式刻画了其 Hamilton 量代数在循环张量积上的作用，建立了公共特征向量与 Drinfeld 多项式因子的对应关系，并证明不可约情形下可用 Bethe ansatz 构造全部特征向量。
We extend the Dong-Mason theorem on the irreducibility of modules for orbifold vertex algebras from [C. Dong, G. Mason, Duke Math. J. 86 (1997)] 305-321] for the category of weak modules. Let $V$ be a vertex operator algebra, $g$ an automorphism of order $p$. Let $W$ be an irreducible weak $V$--module such that $W,W\circ g,\dots,W\circ g^{p-1}$ are inequivalent irreducible modules. We prove that $W$ is an irreducible weak $V^{\left\langle g\right\rangle }$-module. This result can be applied on irreducible modules of certain Lie algebra $\mathfrak L$ such that $W,W\circ g,\dots,W\circ g^{p-1}$ …
Explicit Transgression Primitives on Flag Manifolds
本文在旗流形上构造了显式的相对横gression原语，通过非交换展开和Koszul分解解决了特征形式消失的非构造性问题，并恢复了Poincaré多项式与Euler特征。
Categorification of DAHA and Macdonald polynomials
本文提出与仿射李代数相关的Double Affine Hecke Algebra及其多项式表示和Macdonald多项式的范畴化，通过导出范畴中的函子复合范畴化Demazure算子，并构造模复形使其欧拉示性数等于Macdonald多项式。
A Newton Identity and Finite-Rank Reconstruction for the Queer Lie Superalgebra
本文针对queer李超代数$\frak q_N$建立牛顿型恒等式，推导生成元变换、行列式关系与重构定理，明确中心生成元并刻画一般中心特征。
The Gaudin model for the general linear Lie superalgebra and the completeness of the Bethe ansatz
本文研究一般线性李超代数Gaudin模型，证明奇异空间上的Gaudin代数是循环模和Frobenius代数，并对一般参数证明其可对角化且具有单谱，从而给出Bethe ansatz完备性的新解释。
The Gaudin model and the super Bethe ansatz for unitarizable modules over the general linear Lie (super)algebra
本文研究一般线性李（超）代数 Gaudin 代数在酉模奇异权空间上的作用，证明其为循环模且 Gaudin 代数为 Frobenius 代数，并建立超 Bethe 拟设。
CoHA of Cyclic Quivers and an Integral Form of Affine Yangians
本文研究了循环有向图的CoHA，揭示其与有理Yangian的积分形式的关系，并探讨了相关代数结构与猜想。
Let G be a Lie group with Lie algebra $\mathfrak{g}$, $h \in \frak{g}$ an element for which the derivation ad(h) defines a 3-grading of $\mathfrak{g}$ and $τ_G$ an involutive automorphism of G inducing on $\mathfrak{g}$ the involution $e^{πi ad(h)}$. We consider antiunitary representations $U$ of the Lie group $G_τ= G \rtimes \{e,τ_G\}$ for which the positive cone $C_U = \{ x \in \mathfrak{g} : -i \partial U(x) \geq 0\}$ and $h$ span $\mathfrak{g}$. To a real subspace E of distribution vectors invariant under $exp(\mathbb{R} h)$ and an open subset $O \subseteq G$, we associate the real subspac…
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…
Let $R := R_{2}(p)=\mathbb{C}[t^{\pm 1}, u : u^2 = t(t-α_1)\cdots (t-α_{2n})] $ be the coordinate ring of a nonsingular hyperelliptic curve and let $\mathfrak{g}\otimes R$ be the corresponding current Lie algebra. \color{black} Here $\mathfrak g$ is a finite dimensional simple Lie algebra defined over $\mathbb C$ and \begin{equation*} p(t)= t(t-α_1)\cdots (t-α_{2n})=\sum_{k=1}^{2n+1}a_kt^k. \end{equation*} In earlier work, Cox and Im gave a generator and relations description of the universal central extension of $\mathfrak{g}\otimes R$ in terms of certain families of polynomials $P_{k,i}$ and…
Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and let $λ$ be an arbitrary integral weight. We denote by $\mathbb{B}(λ)$ the crystal of all Lakshmibai-Seshadri paths of shape $λ$. Let $V(λ)$ be the extremal weight module of extremal weight $λ$ generated by the (cyclic) extremal weight vector $v_λ$ of weight $λ$, and let $\mathcal{B}(λ)$ be the crystal basis of $V(λ)$ with $u_λ\in \mathcal{B}(λ)$ the element corresponding to $v_λ$. We prove that the connected component $\mathcal{B}_0(λ)$ of $\mathcal{B}(λ)$ containing $u_λ$ is isomorphic, as a crystal, to the connected compon…
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