From Double Lie Groups to Quantum Groups

On modules for double affine Lie algebras

Imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated, and several results are generalized from the affine Lie algebras. In particular, imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.

Fundamentals of Poisson Lie Groups with Application to the Classical Double

We give a constructive account of the fundamental ingredients of Poisson Lie theory as the basis for a description of the classical double group $D$. The double of a group $G$ has a pointwise decomposition $D\sim G\times G^*$, where $G$ and $G^*$ are Lie subgroups generated by dual Lie algebras which form a Lie bialgebra. The double is an example of a factorisable Poisson Lie group, in the sense of Reshetikhin and Semenov-Tian-Shansky [1], and usually the study of its Poisson structures is developed only in the case when the subgroup $G$ is itself factorisable. We give an explicit description …

Double product integrals and Enriquez quantisation of Lie bialgebras II: The quantum Yang-Baxter equation

For a Lie algebra with Lie bracket got by taking commutators in a nonunital associative algebra L, let T(L) be the vector space of tensors over L equipped with the Ito Hopf algebra structure derived from the associative multiplication in L. We show a necessary and sufficient condition that the double product integral satisfy the quantum Yang-Baxter equation over T(L). We construct a quantisation of an arbitrary quasitriangular Lie bialgebra structure on L in the unital associative subalgebra of T(L)[[h]] consisting of formal power series whose zero order coefficient lies in the space S(L) of s…

On deformed double current algebras for simple Lie algebras

We prove the equivalence of two presentations of deformed double current algebras associated to a complex simple Lie algebra, the first one obtained via a degeneration of affine Yangians while the other one naturally appeared in the construction of the elliptic Casimir connection. We also construct a specific central element of these algebras and, in type A, show that they contain a very large center for certain values of their parameters.

Quantum Double of Yangian of Lie Superalgebra $A(m,n)$ and computation of Universal $R$-matrix

Yangian Double $DY(A(m,n))$ of Lie Superalgebra $A(m,n)$ is described in terms of generators and defining relations. It is proved triangular decomposition for Yangian $Y(A(m,n))$ and its quantum double $DY(A(m,n))$ as a corollary of PBW theorem. It is introduced the normally ordered bases in Yangian and its dual Hopf superalgebra in quantum double. It is calculated the pairing formulas between the elements of its bases. It is received the formula of Universal $R$-matrix of Yangian Double. The formula for Universal $R$-matrix of Yangian, which was introduced by V. Drinfel'd, is received also.

Representations of Double Affine Lie algebras

We study representations of the double affine Lie algebra associated to a simple Lie algebra. We construct a family of indecomposable integrable representations and identify their irreducible quotients. We also give a condition for the indecomposable modules to be irreducible, this is analogous to a result in the representation theory of quantum affine algebras. Finally, in the last section of the paper, we show, by using the notion of fusion product, that our modules are generically reducible.

A stabilizer interpretation of double shuffle Lie algebras

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 …

Poisson Lie Groups, Quantum Duality Principle, and the Quantum Double

The Heisenberg double of a Hopf algebra may be regarded as a quantum analogue of the cotangent bundle of a Lie group. Quantum duality principle describes relations between a Hopf algebra, its dual, and their Heisenberg double in a way which extends both the theory of coadjoint orbits and the classical Fourier transform. We also describe the twisted Heisenberg double which is relevant for the study of nontrivial deformations of the quantized universal enveloping algebras.

关于Kashiwara-Vergne李代数与双洗牌李代数的注记

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。

线性化双 shuffle 李代数的稳定子解释

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。

Courant Algebroids and Strongly Homotopy Lie Algebras

Courant algebroids are structures which include as examples the doubles of Lie bialgebras and the direct sum of tangent and cotangent bundles with the bracket introduced by T. Courant for the study of Dirac structures. Within the category of Courant algebroids one can construct the doubles of Lie bialgebroids, the infinitesimal objects for Poisson groupoids. We show that Courant algebroids can be considered as strongly homotopy Lie algebras.

Bicocycle Double Cross Constructions

We introduce the notion of a bicocycle double cross product (resp. sum) Lie group (resp. Lie algebra), and a bicocycle double cross product bialgebra, generalizing the unified products. On the level of Lie groups the construction yields a Lie group on the product space of two pointed manifolds, none of which being necessarily a subgroup. On the level of Lie algebras, similarly, a Lie algebra is obtained on the direct sum of two vector spaces, none of which is required to be a subalgebra. Finally, on the quantum level the theory presents a bialgebra, on the tensor product of two (co)algebras th…

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。

沿着简单分离曲线拆分戈德曼 - 图拉耶夫李双代数

Splitting the Goldman-Turaev Lie bialgebra along a simple separating curve

研究沿着简单分离曲线拆分戈德曼 - 图拉耶夫李双代数，引入双李双模范畴概念及代数构造，通过几何应用将相关运算组装成双李双模范畴，证明了该李双代数的分解定理。

Bialgebras, the classical Yang-Baxter equation and Manin triples for 3-Lie algebras

This paper studies two types of 3-Lie bialgebras whose compatibility conditions between the multiplication and comultiplication are given by local cocycles and double constructions respectively, and are therefore called the local cocycle 3-Lie bialgebra and the double construction 3-Lie bialgebra. They can be regarded as suitable extensions of the well-known Lie bialgebra in the context of 3-Lie algebras, in two different directions. The local cocycle 3-Lie bialgebra is introduced to extend the connection between Lie bialgebras and the classical Yang-Baxter equation. Its relationship with a te…

Even symplectic supermanifolds and double field theory

Over many decades, the word "double" has appeared in various contexts, at times seemingly unrelated. Several have some relation to mathematical physics. Recently, this has become particularly strking in DFT (double field theory). Two 'doubles' that are particularly relevant are double vector bundles and Drinfel'd doubles. The original Drinfel'd double occurred in the contexts of quantum groups and of Lie bialgebras. Quoting T. Voronov: "Double Lie algebroids arose in the works on double Lie groupoids and in connection with an analog for Lie bialgebroids of the classical Drinfel'd double of Lie…

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。

Double quantization on coadjoint representations of simple Lie groups and its orbits

Let M be a manifold with an action of a Lie group G, $\A$ the function algebra on M. The first problem we consider is to construct a $U_h(\g)$ invariant quantization, $\A_h$, of $\A$, where $U_h(\g)$ is a quantum group corresponding to G. Let s be a G invariant Poisson bracket on M. The second problem we consider is to construct a $U_h(\g)$ invariant two parameter (double) quantization, $\A_{t,h}$, of $\A$ such that $\A_{t,0}$ is a G invariant quantization of $s$. We call $\A_{t,h}$ a $U_h(\g)$ invariant quantization of the Poisson bracket s. In the paper we study the cases when G is a simple …

关于李超代数$\boldsymbol{\frak{gl}_{m|n}}$的双Yangian的量子Berezinian的注记

A note on the quantum Berezinian for the double Yangian of the Lie superalgebra $\mathfrak{gl}_{m|n}$

该研究将量子Berezinian概念推广到李超代数$\frak{gl}_{m|n}$的双Yangian ${\rm DY}(\frak{gl}_{m|n})$，证明其系数是该双Yangian中心的代数独立拓扑生成元。

Topological Lie bialgebra structures and their classification over $ \mathfrak{g}[\![x]\!] $

This paper is devoted to a classification of topological Lie bialgebra structures on the Lie algebra $\mathfrak{g}[\![x]\!]$, where $ \mathfrak{g} $ is a finite-dimensional simple Lie algebra over an algebraically closed field $ F $ of characteristic $ 0 $. We introduce the notion of a topological Manin pair $(L, \mathfrak{g}[\![x]\!])$ and present their classification by relating them to trace extensions of \( F[\![x]\!] \). Then we recall the classification of topological doubles of Lie bialgebra structures on $\mathfrak{g}[\![x]\!]$ and view the latter as a special case of the classificatio…

Contractions on the Classical Double

Lie algebra contractions on the classical Drinfel'd Double of a given Lie bialgebra are introduced and compared to the usual Lie bialgebra contraction theory. The connection between both approaches turns out to be intimately linked to duality problems. The non-relativistic (Galilean) limit of a (1+1) Poincaré Double is used to illustrate the contraction process. Finally, it is shown that, in a certain sense, the classical limit in a quantum algebra can be thought as a certain contraction on the corresponding Double.

李超代数 $\mathfrak{gl}_{m|n}$ 的双 Yangian 与反射代数

Double Yangian and reflection algebras of the Lie superalgebra $\mathfrak{gl}_{m|n}$

本文研究李超代数 $\mathfrak{gl}_{m|n}$ 的双 Yangian，证明其 Poincaré-Birkhoff-Witt 定理、构造量子缩并系数形式的中心元，并据此给出 level 0 反射代数的生成元—关系表述。

Brunnian braids和从双重shuffle李代数到Kashiwara-Vergne李代数的包含

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。

Mould theory and the double shuffle Lie algebra structure

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…

Simple finite-dimensional double algebras

A double algebra is a linear space $V$ equipped with linear map $V\otimes V\to V\otimes V$. Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double algebras do not exist over an arbitrary field, and all simple finite-dimensional associative double algebras over an algebraically closed field are trivial. Over an arbitrary field, every simple finite-dimensional associative double algebra is commutative. A double algebra structure on a finite-dimensional space $V$ is naturally described by a linear operator $…

Unitary spherical representations of Drinfeld doubles

We study irreducible spherical unitary representations of the Drinfeld double of a $q$-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of $SU_q(3)$, we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group $SU_q(2n+1)$ has property (T), which also implies central property (T) of the dual of $SU_q(2n+1)$.

Colour-Kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms

Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contrib…

经典 Lie-Yamaguti Yang-Baxter 方程与 Lie-Yamaguti 双代数

The classical Lie-Yamaguti Yang-Baxter equation and Lie-Yamaguti bialgebras

本文研究 Lie-Yamaguti 双代数，定义经典 Yang-Baxter 方程并建立其与相对 Rota-Baxter 算子的对应，同时证明匹配对、Manin 三元组和双重构造双代数等价。

Generalized cluster structure on the Drinfeld double of $GL_n$

We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group $GL_n$ and derive from it a generalized cluster structure on $GL_n$ compatible with the push-forward of the dual Poisson--Lie bracket.

Drinfeld double of $GL_n$ and generalized cluster structures

We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group $GL_n$ and derive from it a generalized cluster structure on $GL_n$ compatible with the push-forward of the Poisson bracket on the dual Poisson--Lie group.

李超代数$\boldsymbol{\frak{gl}_{M|N}}$的双重杨代数与反射代数，II：量子流

Double Yangian and reflection algebras of the Lie superalgebra $\mathfrak{gl}_{M|N}$, II: Quantum currents

本文研究李超代数$\frak{gl}_{M|N}$的双重杨代数与反射代数，构造临界水平双重杨代数的中心元，推广Etingof-Kazhdan构造引入量子顶点超代数结构，证明反射代数限制模的准模性质。

Manin triples of real simple Lie algebras. Part 1

The article is devoted to the problem of classification of Manin triples up to weak and gauge equivalence. The case of complex simple Lie algebras can be obtained by the papers of A.Belavin, V.Drinfel'd, M.Semenov-Tian-Shanskii. Studing the action of conjugation on complex Manin triples, we get the list of real doubles. We classify all $ad$-invariant forms on the double compatible with bialgebra structure. We classify the Manin triples $(g(R),W,g(C))$ up to weak and gauge equivalence.

双洗牌关系蕴含无穷小六边形方程

Double shuffle relations and double antipodes

本文针对G. Racinet引入的双洗牌李代数$\boldsymbol{\frak{dmr}}_0$，证明其中任意元素均满足无穷小六边形方程，证明方法是比较两个不同的Hopf代数。

Drinfeld Doubles for Finite Subgroups of SU(2) and SU(3) Lie Groups

Drinfeld doubles of finite subgroups of SU(2) and SU(3) are investigated in detail. Their modular data - S, T and fusion matrices - are computed explicitly, and illustrated by means of fusion graphs. This allows us to reexamine certain identities on these tensor product or fusion multiplicities under conjugation of representations that had been discussed in our recent paper [J. Phys. A: Math. Theor. 44 (2011), 295208, 26 pages, arXiv:1103.2943], proved to hold for simple and affine Lie algebras, and found to be generally wrong for finite groups. It is shown here that these identities fail also…

双重Yangian，因子化，以及qKZ方程对于余Lie代数

Double Yangian, Factorization, and qKZ-equation for Cotangent Lie Algebras

本文研究了余Lie代数的双重Yangian及其量子真空模的共形块，证明其满足量子KZ方程，并讨论了与3d $N=4$ gauge theories相关的例子。

Heisenberg double versus deformed derivatives

Two approaches to the tangent space of a noncommutative space whose coordinate algebra is the enveloping algebra of a Lie algebra are known: the Heisenberg double construction and the approach via deformed derivatives, usually defined by procedures involving orderings among noncommutative coordinates or equivalently involving realizations via formal differential operators. In an earlier work, we rephrased the deformed derivative approach introducing certain smash product algebra twisting a semicompleted Weyl algebra. We show here that the Heisenberg double in the Lie algebra case, is isomorphi…

Special symplectic Lie groups and hypersymplectic Lie groups

A special symplectic Lie group is a triple $(G,ω,\nabla)$ such that $G$ is a finite-dimensional real Lie group and $ω$ is a left invariant symplectic form on $G$ which is parallel with respect to a left invariant affine structure $\nabla$. In this paper starting from a special symplectic Lie group we show how to ``deform" the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure $\nabla$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine …

Hom-Lie-Hopf algebras

We studied both the double cross product and the bicrossproduct constructions for the Hom-Hopf algebras of general $(α,β)$-type. This allows us to consider the universal enveloping Hom-Hopf algebras of Hom-Lie algebras, which are of $(α,{\rm Id})$-type. We show that the universal enveloping Hom-Hopf algebras of a matched pair of Hom-Lie algebras form a matched pair of Hom-Hopf algebras. We observe also that, the semi-dualization of a double cross product Hom-Hopf algebra is a bicrossproduct Hom-Hopf algebra. In particular, we apply this result to the universal enveloping Hom-Hopf algebras of a…

3-Lie bialgebras and 3-Lie classical Yang-Baxter equations in low dimensions

In this paper, we give some low-dimensional examples of local cocycle 3-Lie bialgebras and double construction 3-Lie bialgebras which were introduced in the study of the classical Yang-Baxter equation and Manin triples for 3-Lie algebras. We give an explicit and practical formula to compute the skew-symmetric solutions of the 3-Lie classical Yang-Baxter equation (CYBE). As an illustration, we obtain all skew-symmetric solutions of the 3-Lie CYBE in complex 3-Lie algebras of dimension 3 and 4 and then the induced local cocycle 3-Lie bialgebras. On the other hand, we classify the double construc…

多重Eisenstein级数与多重q-zeta值的李代数

Lie algebras for multiple Eisenstein series and multiple q-zeta values

本文证明了多重Eisenstein级数和多重q-zeta值的两个类似李代数猜想，并建立了它们之间的同构关系。

Generalized Bäcklund-Darboux transformations for Coxeter-Toda systems on simple Lie groups

We derive the cluster structure on the conjugation quotient Coxeter double Bruhat cells of a simple Lie group from that on the double Bruhat cells of the corresponding adjoint Lie group given by Fock and Goncharov using the notion of amalgamation given by Fock and Goncharov, and Williams, thereby generalizing the construction developed by Gekhtman \emph{et al}. We will then use this cluster structure on the conjugation quotient Coxeter double Bruhat cells to construct generalized Bäcklund-Darboux transformations between two Coxeter-Toda systems on simple Lie groups in terms of cluster mutation…

Classification of 6-dimensional real Drinfeld doubles

Starting from the classification of real Manin triples done in a previous paper we look for those that are isomorphic as 6-dimensional Lie algebras with the ad-invariant form used for construction of the Manin triples. We use several invariants of the Lie algebras to distinguish the non-isomorphic structures and give explicit form of maps between Manin triples that are decompositions of isomorphic Drinfeld doubles. The result is a complete list of 6-dimensional real Drinfeld doubles. It consists of 22 classes of non-isomorphic Drinfeld doubles.

Lie群与Lie代数

Lie groups and Lie algebras

本文是作者在麻省理工学院讲授的关于Lie群与Lie代数的两个学期课程的扩展笔记。