Denominator identity for affine Lie superalgebras with zero dual Coxeter number
Maria Gorelik, Shifra Reif
证明了对偶Coxeter数为零的非扭仿射李超代数的一个分母恒等式。
We prove a denominator identity for non-twisted affine Lie superalgebras with zero dual Coxeter number.
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Coxeter Graphs for Super Weyl Groups of Exceptional Classical Lie Superalgebras
本文研究例外经典李超代数的基本系统，确定对应超韦尔群的Coxeter图，验证了CLS中关于D(2,1,α)、F(4)和G(3)超韦尔群Coxeter图的猜想。
Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairs
我们提供了一个基本的经典型李超代数的分母和超分母的公式，任何一组正根。我们建立了一个连接的某些集的正根和理论的约化对偶对实李群。作为我们的公式的应用，我们恢复了紧对偶对的Theta对应。沿着这条路，我们给出了基本经典型李超代数的实形式的明确描述。
We prove denominator identities for the periplectic Lie superalgebra $\mathfrak{p}(n)$, thereby completing the problem of finding denominator identities for all simple classical finite-dimensional Lie superalgebras.
Weyl denominator identity for finite-dimensional Lie superalgebras
有限维李超代数的Weyl分母恒等式Kac和M.~ Wakimoto在1994年，被证明。
Denominator Identity for Twisted Affine Lie Superalgebras
李超代数分母恒等式的研究是最近由M. Gorelik，V.G. Kac，P.Moseneder Frajria，I. Musson，P. Papi，M.胁本和作者。本文将这些恒等式推广到扭仿射情形，将结果推广到所有仿射李超代数。
Weyl denominator identity for affine Lie superalgebras with non-zero dual Coxeter number
Kac和Wakimoto给出了Killing型非零的基本李超代数仿射化的Weyl分母恒等式，并在亏一情形下得到了证明.在本文中，我们证明了这个身份。
Weyl denominator identity for the affine Lie superalgebra gl(2|2)^
证明了仿射李超代数gl（2）的Weyl分母恒等式|2）^由V. Kac和M.胁本正如它指出，在他们的论文中，这给了另一个证明雅可比恒等式的数量介绍了一个给定的整数作为一个总和的8平方。
A note on Appell's functions related to the denominators of affine Lie superalgebras $\widehat{sl}(2|1)$ and $\widehat{osp}(3|2)$
本文研究最简 Appell 函数，给出其显式公式、模变换性质和渐近行为，与仿射李超代数分母相关。
We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a $\widehat{sl(2)}$ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of $\widehat{sl(2)}$ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients …
Marked multi-colorings, partially commutative Lie superalgebras and right-angled Coxeter groups
本文利用标记多着色及相关多项式，通过分母恒等式研究 BKM 超代数根重数，并引入 PCLSAs，建立其与直角 Coxeter 群的联系及 Hilbert 级数公式。
The square-root of Siegel modular forms of CHL Z_N orbifolds of type II compactifications are denominator formulae for some Borcherds-Kac-Moody Lie superalgebras for N=1,2,3,4. We study the decomposition of these Siegel modular forms in terms of characters of two sub-algebras: one is a $\widehat{sl(2)}$ and the second is a Borcherds extension of the $\widehat{sl(2)}$. This is a continuation of our previous work where we studied the case of Siegel modular forms appearing in the context of Umbral moonshine. This situation is more intricate and provides us with a new example (for N=5) that did no…
Defining sequences for fundamental root systems and Coxeter graphs for super Weyl groups
本文引入定义序列以描述 A–D 型经典李超代数的基本根系，并利用 Coxeter 图刻画基本经典李超代数超 Weyl 群所对应的 Coxeter 群。
Root groupoid and related Lie superalgebras
本文提出根群胚概念以替代 Kac-Moody 李超代数中的 Weyl 群，构造并分类相关的根超代数，并通过骨架图建立 Coxeter 性质，推广了经典 Weyl 群理论。
We construct a Borcherds Kac-Moody (BKM) superalgebra on which the Conway group Co$_0$ acts faithfully. We show that the BKM algebra is generated by the BRST-closed states in a chiral superstring theory. We use this construction to produce denominator identities for the chiral partition functions of the Conway module $V^{s\natural}$, a supersymmetric $c=12$ chiral conformal field theory whose (twisted) partition functions enjoy moonshine properties and which has automorphism group isomorphic to Co$_0$. In particular, these functions satisfy a genus zero property analogous to that of monstrous …
Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras
设$Γ$是可数交换半群，$A$是满足一定有限性条件的可数交换群.设群$G$作用在$(Γ\times A)$ -分次李超代数${\frak L}=\bigoplus_{(α,a) \in Γ\times A} {\frak L}_{(α,a)}$上，使其保持$(Γ\times A)$ -分次李超代数自同构.本文证明了Euler-Poincaré原理给出了${\frak L}$的广义分母恒等式，并对所有的$g\in G$，$(α,a) \in Γ\times A$导出了超迹$\text{str}(g|{\frak L}_{(α,a)})$的封闭形式公式.讨论了超迹公式在自由李超代数和广义Kac-Moody超代数等各种无限维李超代数中的应用。特别地，我们确定了自由李超代数分解为不可约$GL(n) \times GL(k)$ -模的直和，以及具有群作用的Monstellar李超代数的超迹。最后证明了广义Kac-Moody超代数${\frak g}$上对应于Dynkin图自同构$σ$的Verma模和不可约最高权模的广义特征标与$σ$确定的轨道李超代数$\breve{\frak g}={\frak g}(σ)$上Verma模和不可约最高权模的通常特征标相同.
We study a certain family of finite-dimensional simple representations over quantum affine superalgebras associated to general linear Lie superalgebras, the so-called fundamental representations: the denominators of rational $R$-matrices between two fundamental representations are computed; a cyclicity (and so simplicity) condition on tensor products of fundamental representations is proved.
Integrable highest weight modules over affine superalgebras and number theory
本文的第一部分给出了有限维单李超代数$\frak g\/$的分母恒等式。给出了有限维不可约$\frak g\/$ -模的一个特征标和（超）维数公式。第二部分给出了与$\frak g\/$相关联的仿射超代数$\hat{\frak g}\/$的分母恒等式。专业化的身份给几乎所有的老和许多新的公式的数量表示一个整数的总和平方和总和的三角形数。最后，我们引入了可积$\hat{\frak g}\/$ -模的概念，并给出了不可约可积最高权$\hat{\frak g}\/$ -模的分类。
Kac-Wakimoto character formula for the general linear Lie superalgebra
李超代数的特征标公式在数论和组合数学中有重要的应用。证明了一般线性李超代数gl（m）的Kac-Wakimoto特征标公式|n）。这个公式在模是典型模时是著名的Kac-Weyl特征标公式的特例，在模是平凡模时是Weyl分母恒等式的特例。利用Kac-Wakimoto特征标公式证明了KW-模的行列式特征标公式。
We introduce and study the twisted adapted $r$-cluster point and its combinatorial Auslander-Reiten quivers, called twisted AR-quivers and folded AR-quivers, of type $A_{2n+1}$ which are closely related to twisted Coxeter elements and the non-trivial Dynkin diagram automorphism. As applications of the study, we prove that folded AR-quivers encode crucial information on the representation theory of quantum affine algebra $U_q'(B^{(1)}_{n+1})$ such as Dorey's rule and denominator formulas.
Characters of (relatively) integrable modules over affine Lie superlagebras
本文考虑有限维基本李超代数和仿射李超代数$\mathfrak{g}$上相对可积不可约最高权模$L$的特征标的计算问题。问题由两部分组成。首先，将问题归结为$\overline{\mathfrak{g}}$ -模$F(L)$，其中$\overline{\mathfrak{g}}$是与$L$相关联的整李超代数，$F(L)$是可积不可约最高权$\overline{\mathfrak{g}}$ -模.其次是可积最高权模的特征的计算。关于第一部分有一个普遍的猜想，我们在许多情况下进行了检验。在第二部分中，我们证明了在KW-条件成立的情况下，KW-特征公式在许多情况下都成立，当$\mathfrak{g}$是基本模时，几乎包括所有有限维$\mathfrak{g}$ -模;当$\mathfrak{g}$是仿射非零对偶Coxeter数时，几乎包括所有极大非典型非临界可积$\mathfrak{g}$ -模。
As a continuation of the previous paper, we find a combinatorial interpretation of Dorey's rule for type $C_n$ via twisted Auslander-Reiten quivers (AR-quivers) of type $D_{n+1}$, which are combinatorial AR-quivers related to certain Dynkin diagram automorphisms. Combinatorial properties of twisted AR-quivers are useful to understand not only Dorey's rule but also other notions in the representation theory of the quantum affine algebra $U_q'(C_n^{(1)})$ such as denominator formulas. In addition, unlike twisted adapted classes of type $A_{2n-1}$ in the previous paper, we show twisted AR-quivers…
Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements
本文将重整化Weyl-Kac分母公式改写为仿射Grassmannian元素参数化求和，借助仿射Dynkin图折叠用n-core划分子集基数解释原子长度，并建立钩长统计与Coxeter长度及Nekrasov-Okounkov型公式的联系。
BKM Lie superalgebras from counting twisted CHL dyons
继Sen[arXiv：0911.1563]之后，我们研究了N=4超对称四维CHL模型中有助于扭曲螺旋度迹指数的（“扭曲”）BPS态的计数。用乘法η积给出了半BPS态的生成函数，其中Mathieu群M_{24}起着重要作用。这些乘法的eta产品使我们能够构建西格尔模块化的形式，计数扭曲的四分之一BPS状态。这些Siegel模形式的平方根恰好是Siegel模形式的一个特殊类别，即Clery和Gritsenko[arXiv：0812.3962]所分类的dd-模形式。我们证明了这些dd-模形式中的每一个都是秩为3的Borcherds-Kac-Moody李超代数的Weyl-Kac-Borcherds分母公式。外尔腔的壁与相应的CHL模型中的边缘稳定壁一一对应。这导致了一个周期表的BKM李超代数的性质是一致的物理预期。
In the Coxeter group W(R) generated by the root system R, let T(R) be the number of conjugacy classes having no eigenvalue 1 and let S(R) be the number of conjugacy classes having no eigenvalue -1. The algebra H{R) of observables of the rational Calogero model based on the root system R possesses T(R) independent traces, the same algebra considered as an associative superalgebra with respect to a certain natural parity possesses S(R) even independent supertraces and no odd trace or supertrace. The numbers T(R) and S(R) are determined for all irreducible root systems (hence for all root systems…
In this paper, we introduce twisted and folded AR-quivers of type $A_{2n+1}$, $D_{n+1}$, $E_6$ and $D_4$ associated to (triply) twisted Coxeter elements. Using the quivers of type $A_{2n+1}$ and $D_{n+1}$, we describe the denominator formulas and Dorey's rule for quantum affine algebras $U'_q(B^{(1)}_{n+1})$ and $U'_q(C^{(1)}_{n})$, which are important information of representation theory of quantum affine algebras. More precisely, we can read the denominator formulas for $U'_q(B^{(1)}_{n+1})$ (resp. $U'_q(C^{(1)}_{n})$) using certain statistics on any folded AR-quiver of type $A_{2n+1}$ (resp…
We prove Feigin-Frenkel type dualities between subregular W-algebras of type A, B and principal W-superalgebras of type $\mathfrak{sl}(1|n), \mathfrak{osp}(2|2n)$. The type A case proves a conjecture of Feigin and Semikhatov. Let $(\mathfrak{g}_1,\mathfrak{g}_2) = (\mathfrak{sl}_{n+1},\mathfrak{sl}(1|n+1))$ or $(\mathfrak{so}_{2n+1}, \mathfrak{osp}(2|2n))$ and let $r$ be the lacity of $\mathfrak{g}_1$. Let k be a complex number and $\ell$ defined by $r(k+h^\vee_1)(\ell+h^\vee_2)=1$ with $h^\vee_i$ the dual Coxeter numbers of the $\mathfrak g_i$. Our first main result is that the Heisenberg cos…
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