Showing papers on "Lie bialgebra published in 2016"

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

Abstract: 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 ternary variation of the classical Yang-Baxter equation, called the 3-Lie classical Yang-Baxter equation, a ternary $\mathcal{O}$-operator and a 3-pre-Lie algebra is established. In particular, it is shown that solutions of the 3-Lie classical Yang-Baxter equation give (coboundary) local cocycle 3-Lie bialgebras, whereas 3-pre-Lie algebras give rise to solutions of the 3-Lie classical Yang-Baxter equation. The double construction 3-Lie bialgebra is introduced to extend to the 3-Lie algebra context the connection between Lie bialgebras and double constructions of Lie algebras. Their related Manin triples give a natural construction of pseudo-metric 3-Lie algebras with neutral signature. Moreover, the double construction 3-Lie bialgebra can be regarded as a special class of the local cocycle 3-Lie bialgebra. Explicit examples of double construction 3-Lie bialgebras are provided.

The Frobenius properad is Koszul

Abstract: We show Koszulness of the prop governing involutive Lie bialgebras and also of the props governing non-unital and unital-counital Frobenius algebras, solving a long-standing problem. This gives us minimal models for their deformation complexes, and for deformation complexes of their algebras which are discussed in detail. Using an operad of graph complexes we prove, with the help of an earlier result of one of the authors, that there is a highly non-trivial action of the Grothendieck-Teichm\"uller group $GRT_1$ on (completed versions of) the minimal models of the properads governing Lie bialgebras and involutive Lie bialgebras by automorphisms. As a corollary one obtains a large class of universal deformations of any (involutive) Lie bialgebra and any Frobenius algebra, parameterized by elements of the Grothendieck-Teichm\"uller Lie algebra.We also prove that, for any given homotopy involutive Lie bialgebra structure in a vector space, there is an associated homotopy Batalin-Vilkovisky algebra structure on the associated Chevalley-Eilenberg complex.

Classification of Quantum Groups and Belavin-Drinfeld Cohomologies

Abstract: In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra g. This problem is reduced to the classification of all Lie bialgebra structures on g(K) , where K= C((ħ)). The associated classical double is of the form g(K) ⊗ KA, where A is one of the following: K[ e] , where e2= 0 , K⊕ K or K[ j] , where j2= ħ. The first case is related to quasi-Frobenius Lie algebras. In the second and third cases we introduce a theory of Belavin–Drinfeld cohomology associated to any non-skewsymmetric r-matrix on the Belavin–Drinfeld list (Belavin and Drinfeld in Soviet Sci Rev Sect C: Math Phys Rev 4:93–165, 1984). We prove a one-to-one correspondence between gauge equivalence classes of Lie bialgebra structures on g(K) and cohomology classes (in case II) and twisted cohomology classes (in case III) associated to any non-skewsymmetric r-matrix.

Higher genus Kashiwara-Vergne problems and the Goldman-Turaev Lie bialgebra

Abstract: 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 detail, we show that every solution of ${\rm KV}^{(g,n)}$ induces a Lie bialgebra isomorphism between $\mathfrak{g}^{(g, n+1)}$ and its associated graded ${\rm gr} \, \mathfrak{g}^{(g, n+1)}$. For $g=0$, a similar result was obtained by G. Massuyeau using the Kontsevich integral. This paper is a summary of our results. Details and proofs will appear elsewhere.

Classification of universal formality maps for quantizations of Lie bialgebras

Abstract: 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 formality maps for quantizations of Lie bialgebras are in in 1-1 correspondence with prop morphisms from the minimal resolution AssB_infty of the prop of associative bialgebras to the polydifferential prop DLieB_infty satisfying certain boundary conditions. We prove that the set of such formality morphisms (having an extra property of being Lie connected) is non-empty. The latter result is used in turn to give a short proof of the formality theorem for universal quantizations of arbitrary Lie bialgebras which says that for any Drinfeld associator there is an associated Lie_infty quasi-isomorphism between the Lie_infty algebras controlling, respectively, deformations of the standard bialgebra structure in S(V) and deformations of any given Lie bialgebra structure in V. We study the deformation complex of an arbitrary universal formality morphism and show that it is quasi-isomorphic (up to one class corresponding to the standard rescaling automorphism of the properad LieB) to the full oriented graph complex studied earlier in \cite{Wi2}. This result gives a complete classification of the set of gauge equivalence classes of universal Lie connected formality maps --- it is a torsor over the Grothendieck-Teichmuller group GRT and can hence can be identified with the set of Drinfeld associators.

An explicit two step quantization of Poisson structures and Lie bialgebras

Abstract: We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp. Lie bialgebra) structure. We show explicit transcendental formulae for this correspondence. In the second step one deformation quantizes a quantizable Poisson (resp. Lie bialgebra) structure. We show again explicit transcendental formulae for this second step correspondence (as a byproduct we obtain configuration space models for biassociahedron and bipermutohedron). In the Poisson case the first step is the most non-trivial one and requires a choice of an associator while the second step quantization is essentially unique, it is independent of a choice of an associator and can be done by a trivial induction. We conjecture that similar statements hold true in the case of Lie bialgebras. The main new result is a surprisingly simple explicit universal formula (which uses only smooth differential forms) for universal quantizations of finite-dimensional Lie bialgebras.

M2 to D2 and vice versa by 3-Lie and Lie bialgebra

Abstract: Using the concept of 3-Lie bialgebra, which has recently been defined in arXiv:1604.04475, we construct Bagger-Lambert-Gustavson (BLG) model for M2-brane on Manin triple of a special 3-Lie bialgebra. Then by using the correspondence and relation between those 3-Lie bialgebra with Lie bialgebra, we reduce this model to an $N=(4,4)$ WZW model (D2-brane), such that, its algebraic structure is a Lie bialgebra with one 2-cocycle. In this manner by using correspondence of 3-Lie bialgebra and Lie bialgebra (for this special 3-Lie algebra) one can construct M2-brane from a D2-brane and vice versa.

From Basu-Harvey to Nahm equation via 3-Lie bialgebra

Abstract: Using the concept of 3-Lie bialgebra; we construct the Bagger- Lambert- Gustavson (BLG) model on the Manin triple $\cal D$ of the especial 3-Lie bialgebra $({\cal D},{\cal A}_{\cal G},{\cal A}_{{\cal G}^*}^*)$ which is in correspondence with Manin triple of Lie bialgebra $({\cal D},{\cal G},{\cal G}^*)$. We have shown that the Nahm equation (with Lie bialgebra ${\cal G}$) can be obtained from the Basu-Harvey equation as a boundary condition of BLG model (with 3-Lie bialgebra ${\cal D}$) and vice versa.

M2 to D2 and vice versa by 3-Lie and Lie bialgebra

Abstract: Using the concept of a 3-Lie bialgebra, which has recently been defined in arXiv:1604.04475, we construct a Bagger–Lambert–Gustavson (BLG) model for the M2-brane on a Manin triple of a special 3-Lie bialgebra. Then by using the correspondence and the relation between those 3-Lie bialgebra with Lie bialgebra, we reduce this model to an \(N=(4,4)\) WZW model (D2-brane), such that its algebraic structure is a Lie bialgebra with one 2-cocycle. In this manner by using the correspondence of the 3-Lie bialgebra and Lie bialgebra (for this special 3-Lie algebra) one can construct the M2-brane from a D2-brane and vice versa.

A 2-categorical extension of Etingof-Kazhdan quantisation

Abstract: Let k be a field of characteristic zero. Etingof and Kazhdan constructed a quantisation U_h(b) of any Lie bialgebra b over k, which depends on the choice of an associator Phi. They prove moreover that this quantisation is functorial in b. Remarkably, the quantum group U_h(b) is endowed with a Tannakian equivalence F_b from the braided tensor category of Drinfeld-Yetter modules over b, with deformed associativity constraints given by Phi, to that of Drinfeld-Yetter modules over U_h(b). In this paper, we prove that the equivalence F_b is functorial in b.

On Quantizable Odd Lie Bialgebras

Abstract: Motivated by the obstruction to the deformation quantization of Poisson structures in infinite dimensions, we introduce the notion of a quantizable odd Lie bialgebra. The main result of the paper is a construction of the highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so-called quantizable Poisson structures.

Lie Bialgebras, Fields of Cohomological Dimension at Most 2 and Hilbert's Seventeenth Problem

Abstract: We investigate Lie bialgebra structures on simple Lie algebras of non-split type $A$. It turns out that there are several classes of such Lie bialgebra structures, and it is possible to classify some of them. The classification is obtained using Belavin--Drinfeld cohomology sets, which are introduced in the paper. Our description is particularly detailed over fields of cohomological dimension at most two, and is related to quaternion algebras and the Brauer group. We then extend the results to certain rational function fields over real closed fields via Pfister's theory of quadratic forms and his solution to Hilbert's Seventeenth Problem.

$\mathfrak {g}$-quasi-Frobenius Lie algebras

Abstract: A Lie version of Turaev’s $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a $\mathfrak{g}$-quasi-Frobenius Lie algebra for $\mathfrak{g}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(\mathfrak{q},\beta )$ together with a left $\mathfrak{g}$-module structure which acts on $\mathfrak{q}$ via derivations and for which $\beta $ is $\mathfrak{g}$-invariant. Geometrically, $\mathfrak{g}$-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic Lie groups with an action by a Lie group $G$ which acts via symplectic Lie group automorphisms. In addition to geometry, $\mathfrak{g}$-quasi-Frobenius Lie algebras can also be motivated from the point of view of category theory. Specifically, $\mathfrak{g}$-quasi Frobenius Lie algebras correspond to quasi Frobenius Lie objects in $\mathbf{Rep}(\mathfrak{g})$. If $\mathfrak{g}$ is now equipped with a Lie bialgebra structure, then the categorical formulation of $\overline{G}$-Frobenius algebras given in [16] suggests that the Lie version of a $\overline{G}$-Frobenius algebra is a quasi-Frobenius Lie object in $\mathbf{Rep}(D(\mathfrak{g}))$, where $D(\mathfrak{g})$ is the associated (semiclassical) Drinfeld double. We show that if $\mathfrak{g}$ is a quasitriangular Lie bialgebra, then every $\mathfrak{g}$-quasi-Frobenius Lie algebra has an induced $D(\mathfrak{g})$-action which gives it the structure of a $D(\mathfrak{g})$-quasi-Frobenius Lie algebra.

3-Leibniz bialgebra in $N=6$ Chern-Simons gauge theories, multiple M2 to D2 branes and vice versa

Abstract: Constructing M2-brane and its boundary conditions from D2-brane and the related boundary conditions and vice versa has been possible in our recent work by using 3-Lie bialgebra for BLG model with N = 8 supersymmetry. This could be generalized for BL model with N = 6 by the concept of the 3-Leibniz bialgebra. The 3-Lie bialgebra is an especial case of 3-Leibniz bialgebra, then more comprehensive information will be obtained in this work. Consequently, according to the correspondence of these 3-Leibniz bialgebras with Lie bialgebras, we reduce to D2-brane such that with some restrictions on the gauge field this D2-brane is related to the bosonic sector of an N = (4,4) WZW model equipped with one 2-cocycle in its Lie bialgebra structure. Moreover, the Basu-Harvey equation which is found by considering boundary conditions for BL model containing Leibniz bialgebra structure is reduced to Nahm equation and vice versa using this correspondence.

Lie bialgebra structures on 2-step nilpotent graph algebras

Abstract: We generalize a result on the Heisenberg Lie algebra that gives restrictions to possible Lie bialgebra cobrackets on 2-step nilpotent algebras with some additional properties. For the class of 2-step nilpotent Lie algebras coming from graphs, we describe these extra properties in a very easy graph-combinatorial way. We exhibit applications for $\mathfrak f_n$, the free 2-step nilpotent Lie algebra.

Multiple M2 to D2 and their boundary conditions (and vice versa) via 3-Leibniz bialgebra

Abstract: Using the concept of 3-Leibniz bialgebra, we construct Bagger-Lambert (BL) model for N = 6 multiple M2-brane on Manin triple of special 3-Leibniz bialgebra. Then, according to correspondence of these 3Leibniz bialgebras and Lie bialgebras, we reduce this model to an N = (4,4) WZW model (D2-brane) equipped with one 2-cocycle in it’s Lie bialgebra structure. In other case, the Basu-Harvey equation which is found by considering boundary conditions for BL model containing Leibniz bialgebra structure, reduces to Nahm equation and vice versa using this correspondence. In this way, M2-brane and it’s boundary conditions can be constructed from D2-brane and the related boundary conditions with utility of the correspondence of 3-Leibniz bialgebra and Lie bialgebra, and vice versa.

Lie Bialgebras with Triality and Mal’tsev Bialgebras

Abstract: We consider the relationship between Mal’tsev bialgebras and Lie bialgebras with triality, and also between symplectic Mal’tsev algebras and symplectic Lie algebras with triality. The given relations generalize a connection between Mal’tsev algebras and Lie algebras with triality, revealed by P. O. Mikheev [15], and a connection between Mal’tsev coalgebras and Lie coalgebras with triality, explored by M. E. Goncharov and V. N. Zhelyabin [17].