Let $(g,δ_\hbar)$ be a Lie bialgebra. Let $(U_\hbar(g),Δ_\hbar)$ a quantization of $(g,δ_\hbar)$ through Etingof-Kazhdan functor. We prove the existence of a $L_\infty$-morphism between the Lie algebra $C(\g)=Λ(g)$ and the tensor algebra $TU=T(U_\hbar(g)[-1])$ with Lie algebra structure given by the Gerstenhaber bracket. When $(g,δ_\hbar,r)$ is a coboundary Lie bialgebra, we deduce from the formality morphism the existence of a quantization $R$ of $r$.
In this paper we prove formality of the exterior algebra on V+V* endowed with the big bracket considered as a graded Poisson algebra. We also discuss connection of this result to bialgebra deformations of the symmetric algebra of V considered as bialgebra.
We show that any coboundary Lie bialgebra can be quantized. For this, we prove that: (a) Etingof-Kazhdan quantization functors are compatible with Lie bialgebra twists, and (b) if such a quantization functor corresponds to an even associator, then it is also compatible with the operation of taking coopposites. We also use the relation between the Etingof-Kazhdan construction of quantization functors and the alternative approach to this problem, which was established in a previous work.
We solve a functional version of the problem of twist quantization of a coboundary Lie bialgebra (g,r,Z). We derive from this the following results: (a) the formal Poisson manifolds g^* and G^* are isomorphic; (b) we construct a subalgebra of U(g^*), isomorphic to S(g^*)^g. When g can be quantized, we construct a deformation of the morphism S(g^*)^g subset U(g^*). When g is quasitriangular and nondegenerate, we compare our construction with Semenov-Tian-Shansky's construction of a commutative subalgebra of U(g^*). We also show that the canonical derivation of the function ring of G^* is Hamilt…
We settle several questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. An important new technical ingredient introduced in this paper is an endofunctor D in the category of augmented props with the property that for any representation of a prop P in a vector space V the associated prop DP admits an induced representation on the graded commutative tensor algebra S(V) given in terms of polydifferential operators. Applying this functor to the prop LieB of Lie bialgebras we show that universal formali…
The co-Lie structures compatible with the osp(2|2) Lie super algebra structure are investigated and found to be all of coboundary type. The corresponding classical r-matrices are classified into several disjoint families. The osp(1|2)+u(1) Lie super-bialgebras are also classified.
The first cohomology group of a generalized loop Virasoro algebra with coefficients in the tensor product of its adjoint module is shown to be trivial. The result is applied to prove that Lie bialgebra structures on generalized loop Virasoro algebras are coboundary triangular. We then generalize the results to generalized map Virasoro algebras.
A first goal of this paper is to precisely relate the homotopy theories of bialgebras and $E_2$-algebras. For this, we construct a conservative and fully faithful $\infty$-functor from pointed conilpotent homotopy bialgebras to augmented $E_2$-algebras which consists in an appropriate "cobar" construction. Then we prove that the (derived) formal moduli problem of homotopy bialgebras structures on a bialgebra is equivalent to the (derived) formal moduli problem of $E_2$-algebra structures on this "cobar" construction. We show consequently that the $E_3$-algebra structure on the higher Hochschil…
In a paper by Michaelis a class of infinite-dimensional Lie bialgebras containing the Virasoro algebra was presented. This type of Lie bialgebras was classified by Ng and Taft. In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are classified. It is proved that, for any Lie algebra $W$ of generalized Witt type, all Lie bialgebras on $W$ are coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group $H^1(W,W \otimes W)$ is trivial.
The Lie bialgebras of the (1+1) extended Galilei algebra are obtained and classified into four multiparametric families. Their quantum deformations are obtained, together with the corresponding deformed Casimir operators. For the coboundary cases quantum universal R-matrices are also given. Applications of the quantum extended Galilei algebras to classical integrable systems are explicitly developed.
In this paper, we introduce 3-Hom-Lie bialgebras whose compatibility conditions between the multiplication and comultiplication are given by local cocycle conditions. We study a twisted 3-ary version of the Yang-Baxter Equation, called the 3-Lie classical HomYang-Baxter Equation (3-Lie CHYBE), which is a general form of 3-Lie classical YangBaxter Equation studied in [2] and prove that the bialgebras induced by the solutions of 3-Lie CHYBE induce the coboundary local cocycle 3-Hom-Lie bialgebras.
In this paper, we study Hom-Lie bialgebras by a new notion of the dual representation of a representation of a Hom-Lie algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we introduce the notion of a Hom-Lie bialgebra with emphasis on its compatibility with a Manin triple of Hom-Lie algebras associated to a nondegenerate symmetric bilinear form satisfying a new invariance condition. With this notion, coboundary Hom-Lie bialgebras can be studied without a skew-symmetric condition of $r\in\mathfrak{g}\otimes \mathfrak{g}$, naturally leading to the classical H…
In this paper, Lie bialgebra structures on the extended Schrodinger-Virasoro Lie algebra are classified. It is obtained that all the Lie bialgebra structures on L are triangular coboundary. As a by-product, it is derived that the first cohomology group is trivial.
In this paper, we introduce the notion of a noncommutative Poisson bialgebra, and establish the equivalence between matched pairs, Manin triples and noncommutative Poisson bialgebras. Using quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the introduction of the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra. Rota-Baxter operators, more generally O-operators on noncommut…
In the present paper we shall investigate the Lie bialgebra structures on the Lie algebra $\widetilde{\frak{sl}_2(C_q[x,y])}$, which are shown to be triangular coboundary.
Let $ L $ be a Lie algebra over arbitrary field $ k $ with dim $ L $ =3 and dim $ L' $ =2. All solutions of constant classical Yang- Baxter equation (CYBE) in Lie algebra $ L $ are obtained and the necessary conditions which $ (L,[\ ],Δ_{r}, r) $ is a coboundary (or triangular) Lie bialgebra are given.
All solutions of constant classical Yang-Baxter equation (CYBE) in Lie algebra $L$ with dim $L \le 3$ are obtained and the sufficient and necessary conditions which $(L, \hbox {[ ]}, Δ_r, r)$ is a coboundary (or triangular) Lie bialgebra are given. The strongly symmetric elements in $L\otimes L $ are found and they all are solutions of CYBE in $L$ with $dim L \le 3$.
All coboundary Lie bialgebras and their corresponding Poisson--Lie structures are constructed for the oscillator algebra generated by $\{å,\ap,\am,\bb\}$. Quantum oscillator algebras are derived from these bialgebras by using the Lyakhovsky and Mudrov formalism and, for some cases, quantizations at both algebra and group levels are obtained, including their universal $R$--matrices.
We introduce braided Lie bialgebras as the infinitesimal version of braided groups. They are Lie algebras and Lie coalgebras with the coboundary of the Lie cobracket an infinitesimal braiding. We provide theorems of transmutation, Lie biproduct, bosonisation and double-bosonisation relating braided Lie bialgebras to usual Lie bialgebras. Among the results, the kernel of any split projection of Lie bialgebras is a braided-Lie bialgebra. The Kirillov-Kostant Lie cobracket provides a natural braided-Lie bialgebra on any complex simple Lie algebra $g$, as the transmutation of the Drinfeld-Sklyanin…
All possible Lie bialgebra structures on the harmonic oscillator algebra are explicitly derived and it is shown that all of them are of the coboundary type. A non-standard quantum oscillator is introduced as a quantization of a triangular Lie bialgebra, and a universal $R$-matrix linked to this new quantum algebra is presented.
In this paper, Lie bialgebra structures on generalized Heisenberg-Virasoro algebra $\mathfrak{L}$ are considered. Also, $H^1({\mathfrak{L}} ,\mathfrak{L}\bigotimes\mathfrak{L})$ is given explicitly. Moreover, it is proved that all Lie bialgebra structure on centerless generalized Heisenberg-Virasoro algebra $\bar{\mathfrak{L}}$ are coboundary triangular.
For an oriented 2-dimensional manifold $Σ$ of genus $g$ with $n$ boundary components the space $\mathbb{C}π_1(Σ)/[\mathbb{C}π_1(Σ), \mathbb{C}π_1(Σ)]$ carries the Goldman-Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded (under the natural filtration) is described by cyclic words in $H_1(Σ)$ and carries the structure of a necklace Schedler Lie bialgebra. The isomorphism between these two structures in genus zero has been established in [G. Massuyeau, Formal descriptions of Turaev's loop operations] using Kontsevich integral…
The limiting transitions between different types of quantizations are studied by the deformation theory methods. We prove that for the first order coboundary deformation (g,g*_1 + x g*_2) of a Lie bialgebra (g,g*) one can always get the quantized Lie bialgebra (g,g*_2) as a limit of the sequence of quantizations of the type A(g,g*_1). The obtained results are illustrated by some low-dimensional examples of quantized Lie algebras and superalgebras.
Motivated by recent work on Hom-Lie algebras and the Hom-Yang-Baxter equation, we introduce a twisted generalization of the classical Yang-Baxter equation (CYBE), called the classical Hom-Yang-Baxter equation (CHYBE). We show how an arbitrary solution of the CYBE induces multiple infinite families of solutions of the CHYBE. We also introduce the closely related structure of Hom-Lie bialgebras, which generalize Drinfel'd's Lie bialgebras. In particular, we study the questions of duality and cobracket perturbation and the sub-classes of coboundary and quasi-triangular Hom-Lie bialgebras.
Oriented graph complexes, in which graphs are not allowed to have oriented cycles, govern for example the quantization of Lie bialgebras and infinite dimensional deformation quantization. It is shown that the oriented graph complex GC^or_n is quasi-isomorphic to the ordinary commutative graph complex GC_{n-1}, up to some known classes. This yields in particular a combinatorial description of the action of the Grothendieck-Teichmüller Lie algebra on Lie bialgebras, and shows that a cycle-free formality morphism in the sense of Shoikhet can be constructed rationally without reference to configur…
Let $G$ be a Poisson Lie group and $\g$ its Lie bialgebra. Suppose that $\g$ is a group Lie bialgebra. This means that there is an action of a discrete group $Γ$ on $G$ deforming the Poisson structure into coboundary equivalent ones. Starting from this we construct a non-trivial stack of Hopf-Poisson algebras and prove the existence of associated deformation quantizations. This non-trivial stack is a stack of functions on the formal Poisson group, dual of the starting $Γ$ Poisson-Lie group. To quantize this non-trivial stack we use quantization of a $Γ$ Lie bialgebra which is the infinitesimal…
Lie bialgebra contractions are introduced and classified. A non-degenerate coboundary bialgebra structure is implemented into all pseudo-orthogonal algebras $so(p,q)$ starting from the one corresponding to $so(N+1)$. It allows to introduce a set of Lie bialgebra contractions which leads to Lie bialgebras of quasi-orthogonal algebras. This construction is explicitly given for the cases $N=2,3,4$. All Lie bialgebra contractions studied in this paper define Hopf algebra contractions for the Drinfel'd-Jimbo deformations $U_z so(p,q)$. They are explicitly used to generate new non-semisimple quantum…
Lie bialgebra structures on the extended affine Lie algebra $\widetilde{sl_2(\mathbb{C}_q)}$ are investigated. In particular, all Lie bialgebra structures on $\widetilde{sl_2(\mathbb{C}_q)}$ are shown to be triangular coboundary. This result is obtained by employing some techniques, which may also work for more general extended affine Lie algebras, to prove the triviality of the first cohomology group of $\widetilde{sl_2(\mathbb{C}_q)}$ with coefficients in the tensor product of its adjoint module, namely, $H^1(\widetilde{sl_2(\mathbb{C}_q)},\widetilde{sl_2(\mathbb{C}_q)}\otimes\widetilde{sl_2…
The aim of this paper is to introduce the notion of a mock-Lie bialgebra which is equivalent to a Manin triple of mock-Lie algebras. The study of a special case called coboundary mock-Lie bialgebra leads to the introduction the mock-Lie Yang-Baxter equation on a mock-Lie algebra which is an analogue of the classical Yang-Baxter equation on a Lie algebra. Note that a skew-symmetric solution of mock-Lie Yang-Baxter equation gives a mock-Lie bialgebra. Finally, the notation of $\mathcal O$-operators are studied to construct skew-symmetric solution of mock-Lie Yang-Baxter equation.
All Lie bialgebra structures for the (1+1)-dimensional centrally extended Schrodinger algebra are explicitly derived and proved to be of the coboundary type. Therefore, since all of them come from a classical r-matrix, the complete family of Schrodinger Poisson-Lie groups can be deduced by means of the Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended Galilei and gl(2) Lie bialgebras within the Schrodinger classification are studied. As an application, new quantum (Hopf algebra) deformations of the Schrodinger algebra, including their corresponding quantum universa…
For any integer $d$ we introduce a prop $RHra_d$ of oriented ribbon hypergraphs (in which "edges" can connect more than two vertices) and prove that it admits a canonical morphism of props, $$ Holieb_d^\diamond \longrightarrow RHra_d, $$ $Holieb_d^\diamond$ being the (degree shifted) minimal resolution of prop of involutive Lie bialgebras, which is non-trivial on every generator of $Holieb_d^\diamond$. We obtain two applications of this general construction. As a first application we show that for any graded vector space $W$ equipped with a family of cyclically (skew)symmetric higher products …
Le $X$ be a $C^\infty$-manifold and $\g$ be a finite dimensional Lie algebra acting freely on $X$. Let $r \in \ve^2(\g)$ be such that $Z=[r,r] \in \ve^3(\g)^\g$. In this paper we prove that every quasi-Poisson $(\g,Z)$-manifold can be quantized. This is a generalization of the existence of a twist quantization of coboundary Lie bialgebras (\cite{EH}) in the case $X=G$ (where $G$ is the simply connected Lie group corresponding to $\g$). We deduce our result from a generalized formality theorem. In the case Z=0, we get a new proof of the existence of (equivariant) formality theorem and so (equiv…
We introduce a notion of pre-alternative algebra which may be seen as an alternative algebra whose product can be decomposed into two pieces which are compatible in a certain way. It is also the "alternative" analogue of a dendriform dialgebra or a pre-Lie algebra. The left and right multiplication operators of a pre-alternative algebra give a bimodule structure of the associated alternative algebra. There exists a (coboundary) bialgebra theory for pre-alternative algebras, namely, pre-alternative bialgebras, which exhibits all the familiar properties of the famous Lie bialgebra theory. In par…
Rota-Baxter operators and bialgebras go hand in hand in their applications, such as in the Connes-Kreimer approach to renormalization and the operator approach to the classical Yang-Baxter equation. We establish a bialgebra structure that is compatible with the Rota-Baxter operator, called the Rota-Baxter antisymmetric infinitesimal (ASI) bialgebra. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the context of Rota-Baxter algebras. The study of the coboundary case leads to an enrichment of the associative Yang-B…
In this paper, we construct a bialgebra theory for associative conformal algebras, namely antisymmetric infinitesimal conformal bialgebras. On the one hand, it is an attempt to give conformal structures for antisymmetric infinitesimal bialgebras. On the other hand, under certain conditions, such structures are equivalent to double constructions of Frobenius conformal algebras, which are associative conformal algebras that are decomposed into the direct sum of another associative conformal algebra and its conformal dual as $\mathbb{C}[\partial]$-modules such that both of them are subalgebras an…
All Lie bialgebra structures on the Heisenberg--Weyl algebra $[A_+,A_-]=M$ are classified and explicitly quantized. The complete list of quantum Heisenberg--Weyl algebras so obtained includes new multiparameter deformations, most of them being of the non-coboundary type.
We define a family ${\rm KV}^{(g,n)}$ of Kashiwara-Vergne problems associated with compact connected oriented 2-manifolds of genus $g$ with $n+1$ boundary components. The problem ${\rm KV}^{(0,3)}$ is the classical Kashiwara-Vergne problem from Lie theory. We show the existence of solutions of ${\rm KV}^{(g,n)}$ for arbitrary $g$ and $n$. The key point is the solution of ${\rm KV}^{(1,1)}$ based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman-Turaev Lie bialgebra $\mathfrak{g}^{(g, n+1)}$. In more det…
We establish a bialgebra structure on Rota-Baxter Lie algebras following the Manin triple approach to Lie bialgebras. Explicitly, Rota-Baxter Lie bialgebras are characterized by generalizing matched pairs of Lie algebras and Manin triples of Lie algebras to the context of Rota-Baxter Lie algebras. The coboundary case leads to the introduction of the admissible classical Yang-Baxter equation (CYBE) in Rota-Baxter Lie algebras, for which the antisymmetric solutions give rise to Rota-Baxter Lie bialgebras. The notions of $\mathcal{O}$-operators on Rota-Baxter Lie algebras and Rota-Baxter pre-Lie …
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…
Let $Σ$ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that the solution of generalized (higher genus) Kashiwara-Vergne equations for an automorphism $F \in {\rm Aut}(L)$ of a free Lie algebra implies an isomorphism between the Goldman-Turaev Lie bialgebra $\mathfrak{g}(Σ)$ and its associated graded ${\rm gr}\, \mathfrak{g}(Σ)$. In this paper, we prove the converse: if $F$ induces an isomorphism $\mathfrak{g}(Σ) \cong {\rm gr} \, \mathfrak{g}(Σ)$, then it satisfies the Kashiwara-Vergne equations up to conjugation. As an appli…
We study the problem of classifying all Poisson-Lie structures on the group $G_{\infty}$ of formal diffeomorphisms of the real line $\zR^{1}$ which leave the origin fixed, as well as the extended group of diffeomorphisms $G_{0\infty}\supset G_{\infty}$ whose action on $\zR^{1}$ does not necessarily fix the origin. A complete local classification of all Poisson-Lie structures on the groups $G_{\infty}$ and $G_{0\infty}$ is given. This includes a classification of all Lie-bialgebra structures on the Lie algebra $\Cal G_{\infty}$ of $G_{\infty}$, which we prove to be all of coboundary type, and a…
We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "$S$-equation", which is an analogue of the classical Yang-Baxter equation. In a certain sense, the $S$-equatio…
We construct explicitly a class of coboundary Poisson-Lie structures on the group of formal diffeomorphisms of ${\Bbb R}^n$. Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra $W_n$ of formal vector fields on ${\Bbb R}^n$. We conjecture that this class accounts for all such coboundary structures. The natural action of the constructed Poisson-Lie diffeomorphism groups induces large classes of compatible Poisson structures on ${\Bbb R}^n$, thus making it a Poisson homogeneous space. Moreover, the left-right action of the Poisson-Lie grou…
An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommut…
