Pascal Grosse
A construction of the quantum affine algebra $U_q(g)$ is given in two steps. We explain how to obtain the algebra from its positive Borel subalgebra $U_q(b^+)$, using a construction similar to Drinfeld's quantum double. Then we show how the positive Borel subalgebra can be constructed with quantum shuffles.
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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}.
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。
We express the vanishing conditions satisfied by the correlation functions of Drinfeld currents of quantum affine algebras, imposed by the quantum Serre relations. We discuss the relation of these vanishing conditions with a shuffle algebra description of the algebra of Drinfeld currents.
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。
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.
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 …
Multiplicative analogues of the shuffle elements of the braid group rings are introduced; in local representations they give rise to certain graded associative algebras (b-shuffle algebras). For the Hecke and BMW algebras, the (anti)-symmetrizers have simple expressions in terms of the multiplicative shuffles. The (anti)-symmetrizers can be expressed in terms of the highest multiplicative 1-shuffles (for the Hecke and BMW algebras) and in terms of the highest additive 1-shuffles (for the Hecke algebras). The spectra and multiplicities of eigenvalues of the operators of the multiplication by th…
For a commutative algebra the shuffle product is a morphism of complexes. We generalize this result to the quantum shuffle product, associated to a class of non-commutative algebras (for example all the Hopf algebras). As a first application we show that the Hochschild-Serre identity is the dual statement of our result. In particular, we extend this identity to Hopf algebras. Secondly, we clarify the construction of a class of quasi-Hopf algebras.
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 $(V,m,σ)$.
For any commutative algebra $R$ the shuffle product on the tensor module $T(R)$ can be deformed to a new product. It is called the quasi-shuffle algebra, or stuffle algebra, and denoted $T^q(R)$. We show that if $R$ is the polynomial algebra, then $T^q(R)$ is free for some algebraic structure called Commutative TriDendriform (CTD-algebras). This result is part of a structure theorem for CTD-bialgebras which are associative as coalgebras and whose primitive part is commutative. In other words, there is a good triple of operads $(As, CTD, Com)$ analogous to $(Com, As, Lie)$.
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。
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…
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 $\mathfrak h$.
The quasi-shuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasi-shuffle product algebras as subalgebras of mixable shuffle product algebras. As an application, we obtain Hopf algebra structures in free Rota-Baxter algebras.
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 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.
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.
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 …
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.
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。
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…
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。
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.
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 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.
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.
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…
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…
Rosso and Green have shown how to embed the positive part $U_q(n)$ of a quantum enveloping algebra $U_q(g)$ in a quantum shuffle algebra. In this paper we study some properties of the image of the dual canonical basis $B^*$ of $U_q(n)$ under this embedding $Φ$. This is motivated by the fact that when $g$ is of type $A_r$, the elements of $Φ(B^*)$ are $q$-analogues of irreducible characters of the affine Iwahori-Hecke algebras attached to the groups $GL(m)$ over a $p$-adic field.
The operator valued distributions which arise in quantum field theory on the noncommutative Minkowski space can be symbolized by a generalization of chord diagrams, the dotted chord diagrams. In this framework, the combinatorial aspects of quasiplanar Wick products are understood in terms of the shuffle Hopf algebra of dotted chord diagrams, leading to an algebraic characterization of quasiplanar Wick products as a convolution. Moreover, it is shown that the distributions do not provide a weight system for universal knot invariants.
To any field K of characteristic 0, we associate a set Sha(K). Elements of Sha(K) are equivalence classes of families of Lie polynomials subject to associativity relations. We construct an injection and a retraction between Sha(K) and the set of quantization functors of Lie bialgebras over K. This construction involves the following steps. 1) To each element \varpi of Sha(K), we associate a functor g\mapsto Sh(g) from the category of Lie algebras to that of Hopf algebras; Sh(g) contains Ug. 2) When g and h are Lie algebras, and r_{gh} \in g\otimes h, we construct an element R(r_{gh}) of Sh(g)\…
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。
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。
We summarize the Hopf algebra structure on Feynman diagrams and emphasize the interest in further algebraic structures hidden in Feynman graphs.
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…
Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication * on the set of noncommutative polynomials in A which we call a quasi-shuffle product; it can be viewed as a generalization of the shuffle product. The resulting commutative algebra can be given the structure of a Hopf algebra (_A_,*,Delta). In the case where A is the set of positive integers and the operation on A is addition, (_A_,*,Delta) is the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, there is a Hopf algebra i…
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。
We prove that the Yangian associated to an untwisted symmetric affine Kac-Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed by the authors in arXiv:1407.7994 as an algebraic formalism of the cohomological Hall algebras. As a consequence, we obtain the Poincare-Birkhoff-Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay-Regelskis-Wendlandt.
We present a shuffle realization of the GKLO-type homomorphisms for shifted quantum affine, toroidal, and quiver algebras, thus generalizing its rational version of arXiv:2104.14518 and the type A construction of arXiv:1811.12137. As an application, this allows us to construct large families of commuting and q-commuting difference operators, in particular, providing a convenient approach to the Q-systems where it proves a conjecture of arXiv:1704.00154.
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。
Scary Wheels (and Super Shrubs)
该研究证明了B、C、D型量子仿射超代数的shuffle实现，发展了A类超型灌木理论，定义了对应量子环面超代数，还提出了带参数箭图下shuffle实现的通用框架。
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。
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。
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.
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。
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。
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。
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.
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。
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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