Dual canonical bases, quantum shuffles and q-characters

Generalized virtual braid groups, quasi-shuffle product and quantum groups

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.

分圆双洗牌李代数的另一种 $\mathbb{Q}$-形式

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。

Temperley-Lieb 晶体

Temperley-Lieb Crystals

本文针对 Temperley-Lieb 子式推广 Littlewood-Richardson 法则，引入洗牌 tableau，并利用 Stembridge 公理证明其构成 A 型 Kashiwara 晶体，从而给出相关 Schur 系数的组合解释。

缩减共作用李代数、双洗牌李代数和非交换krv2方程

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。

Quantization of Lie bialgebras and shuffle algebras of Lie algebras

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)\…

The shuffle Hopf algebra and quasiplanar Wick products

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.

A characterization of Nichols algebras of diagonal type which are free algebras

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.

Fox-Neuwirth 胞、量子洗牌代数与 B 型 Artin 群的同调

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。

Spherical Hall algebras of a weighted projective curve

In this article, we deal with the structure of the spherical Hall algebra of coherent sheaves with parabolic structures on a smooth projective curve of arbitrary genus. We provide a shuffle-like presentation of the vector bundle part and show the existence of the generic form. We also prove that the spherical Hall algebra contains the characteristic functions on all the Harder-Narasimhan strata. These results together imply Schiffmann's theorem on the existence of Kac polynomials for quasi-parabolic vector bundles of fixed rank and multi-degree over the curve. On the other hand, the shuffle st…

Quantum invariants for decomposition problems in type A rings of representations

We prove a combinatorial rule for a complete decomposition, in terms of Langlands parameters, for representations of p-adic $GL_n$ that appear as parabolic induction from a large family (ladder representations). Our rule obviates the need for computation of Kazhdan-Lusztig polynomials in these cases, and settles a conjecture posed by Lapid. These results are transferrable into various type A frameworks, such as the decomposition of convolution products of homogeneous KLR-algebra modules, or tensor products of snake modules over quantum affine algebras. The method of proof applies a quantizatio…

循环幂零上同调霍尔代数

The loop-nilpotent cohomological Hall algebra

为具有典范三次势的三倍箭图的循环幂零上同调霍尔代数建立洗牌代数模型，借此关联其与量子库仑分支代数，证明特化后的超交换性，给出两者生成元，刻画BPS李代数，得到Kac多项式新公式并证明相关猜想。

Constructible characters and canonical bases

We give closed formulas for all vectors of the canonical basis of a level 2 irreducible integrable representation of $U_v(sl_\infty)$. These formulas coincide at v=1 with Lusztig's formulas for the constructible characters of the Iwahori-Hecke algebras of type B and D.

An Algebraic Characterization of the Affine Canonical Basis

The canonical basis for quantized universal enveloping algebras associated to the finite--dimensional simple Lie algebras, was introduced by Lusztig. The principal technique is the explicit construction (via the braid group action) of a lattice over $\bz[q^{-1}]$. This allows the algebraic characterization of the canonical basis as a certain bar-invariant basis of $\cl$. Here we present a similar algebraic characterization of the affine canonical basis. Our construction is complicated by the need to introduce basis elements to span the ``imaginary'' subalgebra which is fixed by the affine brai…

Canonical characters on quasi-symmetric functions and bivariate Catalan numbers

Every character on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character (Aguiar, Bergeron, and Sottile, math.CO/0310016). We obtain explicit formulas for the even and odd parts of the universal character on the Hopf algebra of quasi-symmetric functions. They can be described in terms of Legendre's beta function evaluated at half-integers, or in terms of bivariate Catalan numbers: $$ C(m,n)=\frac{(2m)!(2n)!}{m!(m+n)!n!}. $$ Properties of characters and of quasi-symmetric functions are then used to derive several interesting identities among …