Noah Snyder, Peter Tingley
We introduce the notion of a half-ribbon Hopf algebra, which is a ribbon Hopf algebra along with a distinguished element $t$ corresponding to twisting a ribbon by 180 degrees (the half-twist). We show that U_q(g) is a (topological) half-ribbon Hopf algebra, but only if one uses a modified ribbon element. We then discuss some consequences of using this modified ribbon element.
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A module M over a vertex algebra V is half-integrable if a_n act locally nilpotently on M for all a in V, m in M, n>0. We study half-integrable modules over sheaves of twisted chiral differential operators (TCDO) on a smooth variety X. We prove an equivalence between certain categories of half-integrable modules over TCDO and categories of (twisted) D-modules on X.
A half-twist type formula for the R-matrix of a symmetrizable Kac-Moody algebra
Kirillov-Reshetikhin和Levendorskiii-Soibelman发展了一个通用R-矩阵的公式，形式为R=（X^{-1} \otimes X^{-1}）Δ（X）。X在表示V上的作用根据外尔群中的最长元素置换权空间，因此仅定义在有限型中。本文给出了一个类似的公式，该公式适用于任意可对称化Kac-Moody代数的量子化泛包络代数.这是通过将X在V上的作用替换为保持权空间的自同态来实现的，但它是条线性的而不是线性的。
We use elementary skein theory to prove a version of a result of Stylianakis who showed that under mild restrictions on m and n, the normal closure of the m-th power of a half-twist has infinite index in the mapping class group of a sphere with 2n punctures.
Cyclotomic expansions for double twist knots with an odd number of half-twists
本文通过 Kauffman 括号 skein 理论，推导了奇数半扭转双扭结的着色 Jones 多项式分圆展开，解决了 Lovejoy 与 Osburn 在 2019 年提出的问题。
In this paper we generalize Drinfeld's twisted quantum affine algebras to construct twisted quantum algebras for all simply-laced generalized Cartan matrices and present their vertex representation realizations.
We unify and generalize several approaches to constructing braid group representations from finite groups, using iterated twisted tensor products. Our results hint at a relationship between the braidings on the $G$-gaugings of a pointed modular category $\mathcal{C}(A,Q)$ and that of $\mathcal{C}(A,Q)$ itself.
In this paper, we prove the categories of lower bounded twisted modules of positive integer levels for simple vertex operator algebras associated with affine Lie algebras and general automorphisms are semisimple, using the twisted generalization of Zhu's algebra for these vertex operator algebras, constructed in \cite{HY}. We also show that the category of lower bounded twisted modules for a general automorphism is equivalent to the category of lower bounded twisted modules for the corresponding diagram automorphism.
On twisted representations of vertex algebras
在本文中，我们开发了一个形式主义与扭曲实现的顶点和共形代数。作为例子，我们研究了共形代数的扭形式幂级数实现。我们的技术的主要应用是一个非常大的家庭表示的顶点超代数$\goth V_Λ$对应于一个整数格$Λ$的建设。对于由有穷级自同构$σ:Λ\to Λ$生成的自同构$\^σ:\goth V_Λ\to\goth V_Λ$，我们定义了$\goth V_Λ$的扭曲表示范畴$\cal O_{\^σ}$，并证明了这个范畴是半单的，它包含了许多简单对象的同构类.
The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with gl(N). We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the symplectic Lie algebras sp(2n). The representations are parameterized by their highest weights or by their Drinfeld polynomials. In the simplest case of sp(2) we give an explicit description of all the representations as tensor products of evaluation modules. We prove analogues of the Poincare-Birkhoff-Witt theorem for the quantum affine algebra and for the twisted q…
We describe the structure of the irreducible highest weight modules for the twisted Heisenberg-Virasoro Lie algebra at level zero. We prove that such a module is either isomorphic to a Verma module or to a quotient of two Verma modules.
We present a quantization of a Lie coideal structure for twisted half-loop algebras of finite-dimensional simple complex Lie algebras. We obtain algebra closure relations of twisted Yangians in Drinfeld J presentation for all symmetric pairs of simple Lie algebras and for simple twisted even half-loop Lie algebras. We provide the explicit form of the closure relations for twisted Yangians in Drinfeld J presentation for the ${\mathfrak{sl}}_3$ Lie algebra.
We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated to Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu's generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.
Twists of representations of complex reflection groups and rational Cherednik algebras
该研究证明代数分解与诱导表示的上循环扭的一般性结果，将其应用于反射群与有理Cherednik代数，刻画相关模的扭并证明非交换代数的Chevalley定理类似结论。
Representations of the twisted quantized enveloping algebra of type C_n
本文证明了扭曲量子化包络代数U '_q（sp_2 n）的Poincare-Birkhoff-Witt定理的一个版本。这是U_q（gl_2 n）的一个子代数，也是辛李代数的泛包络代数U（sp_2 n）的变形.本文对U '_q（sp_2 n）的有限维不可约表示按其最高权进行了分类，并证明了这些表示是sp_2 n的有限维不可约表示的变形.
We establish the $Q \widetilde{Q}$-systems for the twisted quantum affine algebras that were conjectured in arXiv:1606.05301. We develop the representation theory of Borel subalgebra of twisted quantum affine algebras and we construct their prefundamental representations. We also propose a general conjecture on the relations between twisted and non-twisted types. We prove this conjecture for some particular classes of representations, including prefundamental representations.
Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one defines the notion of a twisted toroidal Lie algebra. In this paper, we construct representations of twisted toroidal Lie algebras from twisted modules over affine and lattice vertex algebras.
A Primer on Twists in the Noncommutative Realm Focusing on Algebra, Representation Theory, and Geometry
本文系统介绍了非交换代数中通过自同构、双代数2-上循环及张量积等方式扭曲乘法结构的几类技术，并侧重通过丰富实例阐明其分类与性质。
We study certain representations of quantum toroidal $\mathfrak{gl}_1$ algebra for $q=t$. We construct explicit bosonization of the Fock modules $\mathcal{F}_u^{(n',n)}$ with a nontrivial slope $n'/n$. As a vector space, it is naturally identified with the basic level 1 representation of affine $\mathfrak{gl}_n$. We also study twisted $W$-algebras of $\mathfrak{sl}_n$ acting on these Fock modules. As an application, we prove the relation on $q$-deformed conformal blocks which was conjectured in the study of $q$-deformation of isomonodromy/CFT correspondence.
We prove that any twisted generalized Weyl algebra satisfying certain consistency conditions can be embedded into a crossed product. We also introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's $q$-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus,…
Twisted regular representations and bimodules in vertex operator algebra theory
本文利用顶点算子代数的扭状正则表示理论，构建扭状Zhu代数上的双模，扩展了Li在非扭情况下的工作，并证实了Dong和Jiang关于双模理论的猜想。
Drinfeld twist for quantum $su(2)$ in the adjoint representation
我们给出了一个详细的描述的伴随表示Drinfeld的扭曲元素，以及其馀产品，为$su_{q}(2)$。作为应用，我们还讨论了该表示中普适R矩阵的计算和具有量子$su(2)$对称性的同粒子态的对称化问题。
Drinfeld Basis And a Nonclassical Free Boson Representation of Twisted Quantum Affine Superalgebra $U_q[osp(2|2)^{(2)}]$}
从超RS代数出发，利用Gauss分解技巧导出了扭量子仿射超代数$U_q[osp(2|2)^{(2)}]$的Drinfeld基.我们用两个$q$变形的自由玻色子场构造了$U_q[osp(2|2)^{(2)}]$的非经典一级表示.
Reconstruction of universal Drinfeld twists from representations
泛德林费尔德扭曲是将量子包络代数的余积与未变形的包络代数的余积联系起来的内自同构。尽管它们支配着经典对称性的变形理论，并且已经出现在许多应用中，但还没有计算过半单量子包络代数的扭曲。有人认为，普遍扭曲可以重建从他们众所周知的表示。给出了一种由包络代数的不可约表示重构包络代数任意元素的方法。对于扭曲，这产生了变形参数中所有阶的代数值生成函数，由基本和普通超几何函数的组合表示。给出了su（2）的三阶万有扭度的显式表达式。
Fusion of twisted representations
推广了手征代数的非扭曲表示的融合积的余乘公式，使之包括任意扭曲表示。我们表明，该公式定义了一个张量积与适当的性质，并确定类似的朱代数任意扭曲表示。作为一个例子，我们研究了N=1和N=2超共形代数的Ramond扇区表示的融合。在后一种情况下，出现了我们详细描述的某些微妙之处。
We study the twist-field representations of W-algebras and generalize construction of the corresponding vertex operators to D- and B-series. It is shown, how the computation of characters of these representations leads to nontrivial identities involving lattice theta-functions. We also propose a way to calculate their exact conformal blocks, expressing them for D-series in terms of geometric data of the corresponding Prym variety for covering curve with involution.
Intertwiners of representations of twisted quantum affine algebras
本文利用q-特征标方法，给出了所有类型扭量子仿射代数第一基本表示R-矩阵的显式计算公式。
Twisted representations of vertex operator algebras
设$V$是顶点算子代数，$g$是有限阶自同构.在$A_g(V)$ -模范畴和某个容许$g$ -扭$V$ -模范畴之间构造了一个结合代数$A_g(V)$和一对函子.特别地，这些函子在每个范畴中的简单模之间表现出双射。我们给出了各种应用，包括这样一个事实，即允许的$g$ -扭曲模的完全约化意味着齐性空间的有限维性和简单的$g$ -扭曲模的数目的有限性。
Finite-dimensional irreducible representations of twisted Yangians
我们研究量子化包络代数称为扭曲杨格。它们类似于B、C和D级数的经典李代数的Yangian Y（gl（N））。扭杨格数是Y（gl（N））中的子代数和关于Y（gl（N））中的余积的余理想。我们给出了它们的有限维不可约表示的完整描述。每一个这样的表示是最高权，我们给出了必要和充分条件，不可约的最高权表示是有限维的。结果类似于德林费尔德定理的普通杨格尔。它的详细证明的A系列也转载。对于最简单的扭曲杨氏，我们构造一个明确的实现，为每个有限维不可约表示在张量积的表示相应的李代数。
We study deformation of tube algebra under twisting of graded monoidal categories. When a tensor category $\mathcal{C}$ is graded over a group $Γ$, a torus-valued 3-cocycle on $Γ$ can be used to deform the associator of $\mathcal{C}$. Based on a natural Fell bundle structure of the tube algebra over the action groupoid of the adjoint action of $Γ$, we show that the tube algebra of the twisted category is a 2-cocycle twisting of the original one.
Twisted $sl(3, {\bf C})^{(2)}_k$ Current Algebra: Free Field Representation and Screening Currents
受扭曲电流代数在描述$Ads_3$黑洞熵中的应用的启发，我们研究了最简单的扭曲电流代数$sl(3,{\bf C})^{(2)}_k$。利用三个$(β,γ)$对和两个标量场，得到了扭曲代数的自由场表示和相应的扭曲Sugawara能量动量张量.初级场和两个第一类屏蔽电流。
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