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 $(Com, As, Lie)$.
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.
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 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 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.
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…
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.
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}.
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.
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…
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.
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 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.
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 …
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…
Folding shuffle algebras and twisted $q$-characters
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Simion Stoilow Institute of Mathematics (IMAR)（罗马尼亚科学院西奥多·斯托伊洛数学研究所）；Faculty of Mathematics, University of Vienna（维也纳大学数学系）
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.
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.
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)\…
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 …
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…
Polylogarithmic characterizations of the reduced coaction Lie algebra and the double shuffle Lie algebra, and their relation to the Kashiwara--Vergne Lie algebra
Université de Genève（日内瓦大学）；Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS（斯特拉斯堡大学与法国国家科学研究中心联合数学高级研究所）
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.
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…
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.
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 …
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.
