Run-Qiang Jian, Marc Rosso, Jiao Zhang
We establish some properties of quantum quasi-shuffle algebras. They include the necessary and sufficient condition for the construction of the quantum quasi-shuffle product, the universal property, and the commutativity condition. As an application, we use the quantum quasi-shuffle product to construct a linear basis of $T(V)$, for a special kind of Yang-Baxter algebras $(V,m,σ)$.
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Using the concept of mixable shuffles, we formulate explicitly the quantum quasi-shuffle product. We also provide a desirable description of the subalgebra generated by the set of primitive elements of the quantum quasi-shuffle bialgebra. A braided coalgebra structure which is dual to the quantum quasi-shuffle in some sense is constructed as well. We use quantum quasi-shuffle algebras to provide examples of Rota-Baxter algebras and tridendriform algebras.
Cohomological properties of the quantum shuffle product and application to the construction of quasi-Hopf algebras
对于一个交换代数，混洗积是复形的一个态射。我们将这个结果推广到量子洗牌积，并与一类非交换代数（例如所有的霍普夫代数）相关联。作为第一个应用程序，我们表明，Hochschild-Serre恒等式是我们的结果的对偶语句。特别地，我们将这个恒等式推广到霍普夫代数。其次，我们阐明了一类拟霍普夫代数的构造。
On correlation functions of Drinfeld currents and shuffle algebras
我们表示的消失条件所满足的相关函数的量子仿射代数的Drinfeld电流，强加的量子塞尔关系。我们讨论的关系，这些消失的条件与洗牌代数描述的代数Drinfeld电流。
Braids, Shuffles and Symmetrizers
乘法类似物的洗牌元素的辫群环的介绍，在当地的代表，他们产生一定的分次结合代数（b-洗牌代数）。对于Hecke代数和BMW代数，（反）对称化子用乘法混洗表示，具有简单的表达式.（反）-对称化子可以用最高乘法1-shuffles（对于Hecke代数和BMW代数）和最高加法1-shuffles（对于Hecke代数）表示。研究了乘性和加性1-shuffle相乘算子的谱和本征值的重性。
On quantum shuffle and quantum affine algebras
分两步给出了量子仿射代数$U_q(g)$的构造。我们解释了如何从它的正Borel子代数$U_q(b^+)$获得代数，使用类似于德林费尔德的量子双。然后，我们展示了如何积极的Borel子代数可以构造量子洗牌。
On the algebra of quasi-shuffles
对于任何交换代数$R$，张量模$T(R)$上的混洗积都可以变形为新的积。它被称为拟混洗代数，或填充代数，并表示为$T^q(R)$。证明了如果$R$是多项式代数，那么对于一个称为交换三角形（CTD-代数）的代数结构，$T^q(R)$是自由的.这个结果是CTD-双代数的结构定理的一部分，CTD-双代数是结合的余代数，其本原部分是交换的。换句话说，存在类似于$(Com, As, Lie)$的良好的三元组操作$(As, CTD, Com)$。
We give shuffle algebra realization of positive part of quantum affine superalgebra $U_{v}(\widehat{\mathfrak{D}}(2,1;θ))$ associated to any simple root systems. We also determine the shuffle algebra associated to $\widehat{\mathfrak{sl}}(2|1)$ with odd root system when $v$ is a primitive root of unity of even order, generalizing results in \cite{FJMMT03}.
Mixable Shuffles, Quasi-shuffles and Hopf Algebras
准混洗积和可混洗积是混洗积的推广，近年来得到了广泛的研究。我们将这两个推广联系起来，并将拟混洗积代数实现为可混洗积代数的子代数。作为应用，我们得到了自由Rota-Baxter代数上的霍普夫代数结构.
The positive part $U^+_q$ of $U_q({\widehat {\mathfrak{sl}}}_2)$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. In 1993 I. Damiani obtained a PBW basis for $U^+_q$, consisting of some elements $\lbrace E_{nδ+α_0}\rbrace_{n=0}^\infty$, $\lbrace E_{nδ+α_1}\rbrace_{n=0}^\infty$, $\lbrace E_{nδ}\rbrace_{n=1}^\infty$ that are defined recursively. Our goal is to describe these elements in closed form. We achieve this goal using Catalan words and a $q$-shuffle algebra.
The real multiple zeta values $ζ(k_1,\ldots,k_r)$ are known to form a ${\bf Q}$-algebra; they satisfy a pair of well-known families of algebraic relations called the double shuffle relations. In order to study the algebraic properties of multiple zeta values, one can replace them by formal symbols $Z(k_1,\ldots,k_r)$ subject only to the double shuffle relations. These form a graded Hopf algebra over ${\bf Q}$, and quotienting this algebra by products, one obtains a vector space. A difficult theorem due to G. Racinet proves that this vector space carries the structure of a Lie coalgebra; in fac…
Using the concept of mixable shuffles, we formulate explicitly the quantum quasi-shuffle product, as well as the subalgebra generated by primitive elements of the quantum quasi-shuffle bialgebra. We construct a braided coalgebra structure which is dual to the quantum quasi-shuffle algebra. We provide representations of quantum quasi-shuffle algebras on commutative braided Rota-Baxter algebras. As an application, we establish a formal power series whose terms come from a special representation of some kind of quasi-shuffle algebra and whose evaluation at 1 is the multiple $q$-zeta values.
In this paper, we introduce certain new features of the shuffle algebra, that will allow us to obtain explicit formulas for the isomorphism between its Drinfeld double and the elliptic Hall algebra.
We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the one defined by [1, 18, 32, 33, 34] using Nakajima quiver varieties.
The Poisson algebra $\mathfrak h$ of invariants of the Nambu-Goto string, which was first introduced by K. Pohlmeyer in 1982, is described using the Shuffle Hopf algebra. In particular, an underlying auxiliary Lie algebra is reformulated in terms of the image of the first Eulerian idempotent of the Shuffle Hopf algebra. This facilitates the comparison of different approaches to the quantization of $\mathfrak h$.
Quasi-shuffle products, introduced by the first author, have been useful in studying multiple zeta values and some of their analogues and generalizations. The second author, together with Kajikawa, Ohno, and Okuda, significantly extended the definition of quasi-shuffle algebras so it could be applied to multiple zeta q-values. This article extends some of the algebraic machinery of the first author's original paper to the more general definition, and uses this extension to obtain various algebraic formulas in the quasi-shuffle algebra in a transparent way. Some applications to multiple zeta va…
Quasi-modularity of symmetric quasi-shuffles
本文建立代数框架，研究对称多重 $q$-zeta 值的拟模性，识别出穷尽 level-one 拟模形式及有限 level 拟模形式的对称拟混洗类，并给出对称化公式及与对称函数的联系。
In this paper generalized Matsumoto-Tits sections lifting permutations to the algebra associated to a generalized virtual braid monoid are defined. They are then applied to study the defining relations of the quantum quasi-shuffle algebras via the total symmetrization operator.
We prove that the double shuffle Lie algebra ds, dual to the space of new formal multiple zeta values, injects into the Kashiwara-Vergne Lie algebra krv defined and studied by Alekseev-Torossian. The proof is based on a reformulation of the definition of krv, and uses a theorem of Ecalle on a property of elements of ds.
Shuffle algebra realizations for modular Yangians
本文研究了在特征p>2的域上，sl_n的杨代数正子代数Y_n^(>)(k)的洗牌代数实现，确定了其核和像，并揭示了与特征p相关的轮条件。
We introduce in this paper the generalized virtual braid group on n strands GVB_n, generalizing simultaneously the braid groups and their virtual versions. A Mastumoto-Tits type section lifting shuffles in a symmetric group S_n to the monoid associated to GVB_n is constructed, which is then applied to characterize the quantum quasi-shuffle product. A family of representations of GVB_n is constructed using quantum groups.
In this letter, we use quantum quasi-shuffle algebras to construct Rota-Baxter algebras, as well as tridendriform algebras. We also propose the notion of braided Rota-Baxter algebras, which is the relevant object of Rota-Baxter algebras in a braided tensor category. Examples of such new algebras are provided by using quantum multi-brace algebras in a category of Yetter-Drinfeld modules.
The shuffle Hopf algebra and quasiplanar Wick products
在非对易闵可夫斯基空间上的量子场论中出现的算符值分布可以用弦图的推广，即点弦图来表示。在这个框架中，准平面威克产品的组合方面的理解，在洗牌霍普夫代数的虚线弦图，导致准平面威克产品作为卷积的代数表征。此外，它示出的分布不提供一个通用的结不变量的重量系统。
According to Racinet's work, the scheme of double shuffle and regularization relations between cyclotomic analogues of multiple zeta values has the structure of a torsor over a pro-unipotent $\mathbb Q$-algebraic group $\sf{DMR}_0$, which is an algebraic subgroup of a pro-unipotent $\mathbb Q$-algebraic group of outer automorphisms of a free Lie algebra. We show that the harmonic (stuffle) coproduct of double shuffle theory may be viewed as an element of a module over the above group, and that $\sf{DMR}_0$ identifies with the stabilizer of this element. We identify the tangent space at origin …
A tale of two shuffle algebras
本文从量子仿射群赋值表示与R矩阵出发，给出量子环面代数“上半”“下半”部分的洗牌代数表示，得到可延拓水平量子仿射子代数Drinfeld-Jimbo余积的新拓扑余积。
Quasi-shuffle products
给定一个局部有限分次集A和A上一个增加次数的交换结合运算，我们在A中的非交换多项式集上构造一个交换乘法 *，我们称之为拟混洗积;它可以被看作是混洗积的推广。所得到的交换代数可以给出一个霍普夫代数（_A_，*，Delta）的结构.在A是正整数的集合且A上的运算是加法的情况下，（_A_，*，Delta）是拟对称函数的霍普夫代数.如果允许有理系数，则存在从A上的shuffle霍普夫代数到（_A_，*，Delta）的霍普夫代数同构exp.我们讨论了（_A_，*，Delta）的对偶，定义了一个变形 *_q，当q = 1时，它与 * 重合，当q不是单位根时，它与级联积同构.最后，我们讨论了这种结构的各种例子。
A quantum shuffle approach to quantum affine super algebra of type $C(2)^{(2)}$ and its equitable presentation
本文提出利用量子洗牌方法和卡塔兰词，以闭式表达式呈现类型 $C(2)^{(2)}$ 的量子拟叠代数的两个基，并给出了其玻色化表示。
Folding shuffle algebras and twisted $q$-characters
通过引入折叠洗牌代数，证明了量子非扭曲仿射代数模的某些 $q$-特征与其扭曲对应物相等的Hernandez猜想，并将结果推广到具有自同构的任意箭图，特别地定义并描述了扭曲量子环面代数。
An alternative $\mathbb{Q}$-form of the cyclotomic double shuffle Lie algebra
本文为 Racinet 分圆双洗牌李代数提出了一种受同余多重 zeta 值双洗牌关系启发的替代 $\mathbb{Q}$-形式，并建立不变性定理证明两者在 Galois 作用下可相互重构。
In this paper, we prove that the quantum toroidal algebra of gl_n is isomorphic to the double shuffle algebra of Feigin and Odesskii for the cyclic quiver. The shuffle algebra viewpoint will allow us to prove a factorization formula for the universal R-matrix of the quantum toroidal algebra.
Isomorphism between Hopf algebras for multiple zeta values
本文证明了多重ζ值的准洗牌代数与洗牌代数通过准对称函数具有霍普夫代数同构性，并与已知的霍夫曼等人同构性进行了比较。
We prove in this paper a Borel-Weil-Bott type theorem for the coHochschild homology of a quantum shuffle algebra associated with quantum group datum taking coefficients in some well-chosen bicomodules, which can be looked as an analogue of equivariant line bundles over flag varieties in the non-commutative case.
Rota-Baxter algebras and the closely related dendriform algebras have important physics applications, especially to renormalization of quantum field theory. Braided structures provide effective ways of quantization such as for quantum groups. Continuing recent study relating the two structures, this paper considers Rota-Baxter algebras and dendriform algebras in the braided contexts. Applying the quantum shuffle and quantum quasi-shuffle products, we construct free objects in the categories of braided Rota-Baxter algebras and braided dendriform algebras, under the commutativity condition. We f…
Let $\bigwedge_σV=\bigoplus_{k\geq 0}\bigwedge_σ^kV$ be the quantum exterior algebra associated to a finite-dimensional braided vector space $(V,σ)$. For an algebra $\mathfrak{A}$, we consider the convolution product on the graded space $\bigoplus_{k\geq 0}\mathrm{Hom}\big(\bigwedge_σ^kV,\bigwedge_σ^kV\otimes \mathfrak{A}\big)$. Using this product, we define a notion of quantum minor determinant of a map from $V$ to $V\otimes \mathfrak{A}$, which coincides with the classical one in the case that $\mathfrak{A}$ is the FRT algebra corresponding to $U_q(\mathfrak{sl}_N)$. We establish quantum Lap…
We relate shuffle algebras, as defined by Nichols, Feigin-Odesskii and Rosso, to perverse sheaves on symmetric products of the complex line (i.e., on the spaces of monic polynomials stratified by multiplicities of roots). More precisely, we construct an equivalence between: (i) Braided Hopf algebras of a certain type. (ii) Factorizable collections of perverse sheaves on all the symmetric products. Under this eqiuvalence, the Nichols algebra associated to an object V corresponds to the collection of the intersection cohomology extensions of the local systems on the open configuration spaces ass…
This paper is devoted to explore the freeness of Nichols algebras of diagonal type and to determine the dimension of the kernel of the shuffle map considered as an operator acting on the free algebra. Our proof is based on an inequality for the number of Lyndon words and on an identity for the shuffle map. For a particular family of examples, the freeness of the Nichols algebra is characterized in terms of solutions of a quadratic diophantine equation.
Notes on Kashiwara-Vergne and double shuffle Lie algebras
本文梳理了Kashiwara-Vergne李代数$\frak{krv}$与双洗牌李代数$\frak{dmr}$的关系研究进展，并验证了Ecalle六元关系在小深度情形下的有效性。
Quantization of Lie bialgebras and shuffle algebras of Lie algebras
对于任何特征为0的域K，我们关联一个集合Sha（K）。Sha（K）的元素是具有结合关系的李多项式族的等价类。我们构造了Sha（K）与K上李双代数的量子化函子集之间的一个注入和一个收缩。该构造涉及以下步骤。1)对于Sha（K）的每个元素\varpi，我们将一个函子g \mapsto Sh（g）从李代数的范畴关联到霍普夫代数的范畴; Sh（g）包含Ug. 2)当g和h是李代数，r_{gh} \in g \otimes h时，我们构造了Sh（g）\otimes Sh（h）的一个满足拟三角恒等式的元素R（r_{gh}），R（r_{gh}）定义了一个从Sh（g）^* 到Sh（h）的霍普夫代数态射. 3)当g = h且r \in g \otimes g是CYBE的解时，我们构造了一个级数ρ（r），使得R（ρ（r））是QYBE的解.ρ（r）关于r的表达式涉及李多项式，并且我们证明了这个表达式在普适水平上是唯一的.这一步依赖于消失声明的上同调所产生的泛代数的解决方案的CYBE。4)我们定义李双代数g的量子化为R（ρ（r））定义的态射的像，其中r \in g \otimes g^* 是附着在g上的标准元。
Shuffle Products, Degenerate Affine Hecke Algebras, and Quantum Toda Lattice
本文从代数角度重新识别GL_N的量子Toda格子与截断移位Yangian，并通过有限Miura变换和mirabolic约简方法提出新的代数构造。
Let $\mathbb F$ denote a field, and pick a nonzero $q \in \mathbb F$ that is not a root of unity. Let $\mathbb Z_4=\mathbb Z/4 \mathbb Z$ denote the cyclic group of order 4. Define a unital associative ${\mathbb F}$-algebra $\square_q$ by generators $\lbrace x_i \rbrace_{i \in \mathbb Z_4}$ and relations $$\frac{q x_i x_{i+1}-q^{-1}x_{i+1}x_i}{q-q^{-1}} = 1,\qquad x^3_i x_{i+2} - \lbrack 3 \rbrack_q x^2_i x_{i+2} x_i + \lbrack 3 \rbrack_q x_i x_{i+2} x^2_i -x_{i+2} x^3_i = 0,$$ where $\lbrack 3 \rbrack_q = \big(q^3-q^{-3}\big)/\big(q-q^{-1}\big)$. Let $V$ denote a $\square_q$-module. A vector …
Fusion and specialization for type ADE shuffle algebras
本文研究了ADE类型中混洗代数的融合与专门化，通过凸序构建融合电流并分析其与根向量的关系。
A stabilizer interpretation of the (extended) linearized double shuffle Lie algebra
本文通过稳定子解释方法，将线性化双 shuffle 李代数及其扩展与多重 zeta 值和多重 q-zeta 值联系起来，揭示了稳定子在代数结构中的作用。
We show that the shuffle algebra associated to a doubled quiver (determined by 3-variable wheel conditions) is generated by elements of minimal degree. Together with results of Varagnolo-Vasserot and Yu Zhao, this implies that the aforementioned shuffle algebra is isomorphic to the localized K-theoretic Hall algebra associated to the quiver by Schiffmann-Vasserot. With small modifications, our theorems also hold under certain specializations of the equivariant parameters, which will allow us in Neguţ-Sala-Schiffmann to give a generators-and-relations description of the Hall algebra of any curv…
Feynman diagrams and polylogarithms: shuffles and pentagons
我们总结了费曼图上的Hopf代数结构，并强调对费曼图中隐藏的进一步代数结构的兴趣。
BPS Lie algebras, perverse filtrations and shuffle algebras
通过将上同调Hall代数上的反常滤过与多项式的极限条件关联，显式描述了零势能箭图的BPS李代数，并部分推广到任意势能情形。
Shuffle algebras, lattice paths and quantum toroidal $\mathfrak{gl}_{n|m}$
本文研究了$\mathfrak{gl}_{n|m}$矩阵洗牌代数中的通交元素，通过量子 affine 代数$U_t(\dot{\mathfrak{gl}}_{n|m})$的$R$-矩阵部分迹给出公式，并赋予格路解释。
Polylogarithmic characterizations of the reduced coaction Lie algebra and the double shuffle Lie algebra, and their relation to the Kashiwara--Vergne Lie algebra
本文研究了缩减共作用李代数与双洗牌李代数之间的关系，并展示了非交换krv2方程的构造方法。
Dual canonical bases, quantum shuffles and q-characters
Rosso和Green已经展示了如何将量子包络代数$U_q(g)$的正部分$U_q(n)$嵌入到量子混洗代数中。本文研究了$U_q(n)$的对偶正则基$B^*$在这种嵌入$Φ$下的象的一些性质。这是由于当$g$是$A_r$类型时，$Φ(B^*)$的元素是附着在$p$ -进域上的群$GL(m)$上的仿射Iwahori-Hecke代数的不可约特征标的$q$ -模拟。
Brunnian braids and the inclusion from double shuffle Lie algebra to Kashiwara-Vergne Lie algebra
本文证明了Brunnian braids的双重shuffle李代数通过infinitesimal hexagon方程注入到对称Kashiwara-Vergne李代数，并扩展了这一包含到两个方向：利用Brunnian Lie代数的下正则系列以及建立pentagon方程映射、stuffle coproduct、divergence映射和necklace cobracket之间的显式联系。
We provide a method to construct new associators out of Drinfel'd's KZ associator. We obtain two analytic families of associators whose coefficients we can describe explicitly by a generalization of multiple zeta values. The two families contain two different paths that deform the Drinfel'd KZ associator into the trivial associator 1. We show that both paths are injective, that is, all of the associators parametrized by them are different. Our construction is based on the observation that one can recover multiple polylogarithms as generating functions of order polynomials of certain formally c…
Shuffle algebras and their integral forms: specialization map approach in types $C_n$ and $D_n$
针对Cₙ和Dₙ型量子环代数正子代数及其整形式，通过新Drinfeld实现构造PBWD基，建立洗牌代数实现并推广到整形式，自然推广经典型处理且特殊化映射更优。
The purpose of this paper is to prove the upper bound in Malle's conjecture on the distribution of finite extensions of $\mathbb{F}_q(t)$ with specified Galois group. As in previous work of Ellenberg-Venkatesh-Westerland, our result is based upon computations of the homology of braid groups with certain (exponential) coefficients. However, the approach in this paper is new, relying on a connection between the cohomology of Hurwitz spaces and the cohomology of quantum shuffle algebras.
Using Catalan words and a $q$-shuffle algebra to describe the Beck PBW basis for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$
本文利用Catalan词和$q$-shuffle代数，以闭式表达量子包络代数正部的Beck PBW基，并探讨其与Damiani PBW基及交替PBW基的关系。
Specialization maps for shuffle algebras of type $B_{n}$ and $G_{2}$
本研究通过特殊化映射构建$B_n$型与$G_2$型Feigin-Odesskii洗牌代数的滤过，推广了$A_n$型相关结论，且该滤过与对应类型量子流代数的PBW型基相容。
Fox-Neuwirth cells, quantum shuffle algebras, and the homology of type-B Artin groups
本文从辫子向量空间发展 B 型 Artin 群的辫子表示，将其同调与量子洗牌代数上双模的上同调联系起来，并完整刻画了特征 0 域上一维辫子表示系数下的同调。
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 …
Weight filtration of Hurwitz spaces and quantum shuffle algebras
本文通过建立 Hurwitz 空间与量子洗牌代数的重量滤波等价关系，揭示了 Hurwitz 空间权重的显式上界。
Fox-Neuwirth cells, quantum shuffle algebras, and character sums of the resultant
本文利用拓扑方法计算多重穿孔平面辫群同调并证明混合辫群同调的消没范围，从而给出给定次数首一无平方因子多项式对上结式特征和的上界，回答了 Ellenberg 和 Shusterman 在二次情形下的问题。
We compute the homology of the complex of formal operations on the Hochschild complex of differential graded commutative algebras as defined by Wahl and prove that these can be built as infinite sums of operations obtained from Loday's shuffle operations, Connes' boundary operator and the shuffle product.
Quantum loop groups and shuffle algebras via Lyndon words
本文通过环路词组合学推广有限型方法，研究量子环路群的PBW基，并证明Enriquez同态到椭圆代数三角退化为同构。
Using a $q$-shuffle algebra to describe the basic module $V(Λ_0)$ for the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$
本文利用 $q$-shuffle 代数构造量子包络代数 $U_q(\widehat{\mathfrak{sl}}_2)$ 的基本模 $V(Λ_0)$，证明其与由空字生成的子模同构，并给出该子模的字空间刻画。
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