Lie Bialgebra Structures Research Articles

Quantization of the Rank Two Heisenberg–Virasoro Algebra

Quantum groups occupy a significant position in both mathematics and physics, contributing to progress in these fields. It is interesting to obtain new quantum groups by the quantization of Lie bialgebras. In this paper, the quantization of the rank two Heisenberg–Virasoro algebra by Drinfel’d twists is presented, Lie bialgebra structures of which have been investigated by the authors recently.

Integrable deformations of Rikitake systems, Lie bialgebras and bi-Hamiltonian structures

Integrable deformations of a class of Rikitake dynamical systems are constructed by deforming their underlying Lie–Poisson Hamiltonian structures, which are considered linearizations of Poisson–Lie structures on certain (dual) Lie groups. By taking into account that there exists a one-to one correspondence between Poisson–Lie groups and Lie bialgebra structures, a number of deformed Poisson coalgebras can be obtained, which allow the construction of integrable deformations of coupled Rikitake systems. Moreover, the integrals of the motion for these coupled systems can be explicitly obtained by means of the deformed coproduct map. The same procedure can be also applied when the initial system is bi-Hamiltonian with respect to two different Lie–Poisson algebras. In this case, to preserve a bi-Hamiltonian structure under deformation, a common Lie bialgebra structure for the two Lie–Poisson structures has to be found. Coupled dynamical systems arising from this bi-Hamiltonian deformation scheme are also presented, and the use of collective ‘cluster variables’, turns out to be enlightening in order to analyse their dynamical behaviour. As a general feature, the approach here presented provides a novel connection between Lie bialgebras and integrable dynamical systems.

1-Cocycles of the Witt algebra with coefficients in tensor product of modules

In this paper, we classify 1-cocycles of the Witt algebra with coefficients in the tensor product of two arbitrary tensor density modules. In a special case, we recover a theorem originally established by Ng and Taft in [24]. Furthermore, by these 1-cocycles, we determine Lie bialgebra structures over certain infinite-dimensional Lie algebras containing the Witt algebra.

Classification of D-bialgebra structures on power series algebras

In this paper, we use algebro-geometric methods in order to derive classification results for so-called D-bialgebra structures on the power series algebra A〚z〛 for certain central simple non-associative algebras A. These structures are closely related to a version of the classical Yang-Baxter equation (CYBE) over A.If A is a Lie algebra, we obtain new proofs for pivotal steps in the known classification of non-degenerate topological Lie bialgebra structures on A〚z〛 as well as of non-degenerate solutions of the usual CYBE.If A is associative, we achieve the classification of non-triangular topological balanced infinitesimal bialgebra structures on A〚z〛 as well as of all non-degenerate solutions of an associative version of the CYBE.

Topological Manin pairs and (n,s)-type series

Lie subalgebras of L = {mathfrak {g}}(!(x)!) times {mathfrak {g}}[x]/x^n{mathfrak {g}}[x] , complementary to the diagonal embedding Delta of {mathfrak {g}}[![x]!] and Lagrangian with respect to some particular form, are in bijection with formal classical r-matrices and topological Lie bialgebra structures on the Lie algebra of formal power series {mathfrak {g}}[![x]!] . In this work we consider arbitrary subspaces of L complementary to Delta and associate them with so-called series of type (n, s) . We prove that Lagrangian subspaces are in bijection with skew-symmetric (n, s) -type series and topological quasi-Lie bialgebra structures on {mathfrak {g}}[![x]!] . Using the classificaiton of Manin pairs we classify up to twisting and coordinate transformations all quasi-Lie bialgebra structures. Series of type (n, s) , solving the generalized classical Yang-Baxter equation, correspond to subalgebras of L. We discuss their possible utility in the theory of integrable systems.

Lie algebras of curves and loop-bundles on surfaces

W. Goldman and V. Turaev defined a Lie bialgebra structure on the $$\mathbb Z$$ -module generated by free homotopy classes of loops of an oriented surface (i.e. the conjugacy classes of its fundamental group). We generalize this construction to a much larger space of equivalence classes of curves by replacing homotopies by thin homotopies, following the combinatorial approach of M. Chas. As an application we use properties of the generalized bracket to give a geometric proof of a conjecture by Chas in the original setting of full homotopy classes, namely a characterization of homotopy classes of simple curves in terms of the Goldman–Turaev bracket.

Quantization of the derivation Lie algebra over quantum torus

Recently, Lie bialgebra structures of the derivation Lie algebra over quantum torus were determined. In this paper, we use the general method of quantization by a Drinfel’d twist to quantize these algebras with their Lie bialgebra structures and present the explicit formulae.

Lie Bialgebras on the Rank Two Heisenberg–Virasoro Algebra

The rank two Heisenberg–Virasoro algebra can be viewed as a generalization of the twisted Heisenberg–Virasoro algebra. Lie bialgebras play an important role in searching for solutions of quantum Yang–Baxter equations. It is interesting to study the Lie bialgebra structures on the rank two Heisenberg–Virasoro algebra. Since the Lie brackets of rank two Heisenberg–Virasoro algebra are different from that of the twisted Heisenberg–Virasoro algebra and Virasoro-like algebras, and there are inner derivations (from itself to its tensor space) which are hidden more deeply in its interior algebraic structure, some new techniques and strategies are employed in this paper. It is proved that every Lie bialgebra structure on the rank two Heisenberg–Virasoro algebra is triangular coboundary.

Lie Bialgebra Structures on the Lie Algebra L Related to the Virasoro Algebra

A Lie bialgebra is a vector space endowed simultaneously with the structure of a Lie algebra and the structure of a Lie coalgebra, and some compatibility condition. Moreover, Lie brackets have skew symmetry. Because of the close relation between Lie bialgebras and quantum groups, it is interesting to consider the Lie bialgebra structures on the Lie algebra L related to the Virasoro algebra. In this paper, the Lie bialgebras on L are investigated by computing Der(L, L⊗L). It is proved that all such Lie bialgebras are triangular coboundary, and the first cohomology group H1(L, L⊗L) is trivial.

Conformal Yang–Baxter equation on Cur(sl2(C))

In 2008, Liberati [J. Algebra 319, 2295–2318 (2008)] defined what a conformal Lie bialgebra is and introduced the conformal classical Yang–Baxter equation (CCYBE). An L-invariant solution to the weak version of CCYBE provides a conformal Lie bialgebra structure. We describe all solutions to the CCYBE on the current Lie conformal algebra Cur(sl2(C)) and to the weak version of it.

Dual Lie bialgebra structures of the twisted Heisenberg–Virasoro type

In this paper, by studying the maximal good subspaces, we determine the dual Lie coalgebras of the centerless twisted Heisenberg–Virasoro algebra. Based on this, we construct the dual Lie bialgebras structures of the twisted Heisenberg–Virasoro type. As by-products, four new infinite dimensional Lie algebras are obtained.

Prop of Ribbon Hypergraphs and Strongly Homotopy Involutive Lie Bialgebras

Abstract For any integer $d$, we introduce a prop ${{\mathcal {R}}}{{\mathcal {H}}} ra_d$ of oriented ribbon hypergraphs (in which “edges” can connect more than two vertices) and prove that it admits a canonical morphism of props, $$ \begin{align*} & {\mathcal{H}}\textit{olieb}_{d}^\diamond \longrightarrow {{\mathcal{R}}}{{\mathcal{H}}} ra_d, \end{align*}$$${\mathcal {H}}\textit {olieb}_{d}^\diamond $ being the (degree shifted) minimal resolution of prop of involutive Lie bialgebras, which is nontrivial on every generator of ${\mathcal {H}}\textit {olieb}_{d}^\diamond $. We obtain two applications of this general construction. First we show that for any graded vector space $W$ equipped with a family of cyclically (skew)symmetric higher products, $$ \begin{align*} & \Theta_n: (\otimes^n W[d])_{{{\mathbb{Z}}}_n} \longrightarrow {{\mathbb{K}}}[1+d], \ \ \ \ \ n\geq 1, \end{align*}$$the associated vector space of cyclic words $Cyc(W)=\oplus _{n\geq 0} (\otimes ^n W)_{{{\mathbb {Z}}}_n} $ has a combinatorial ${\mathcal {H}}\textit {olieb}_{d}^\diamond $-structure. As an illustration, we construct for each natural number $N\geq 1$ an explicit combinatorial strongly homotopy involutive Lie bialgebra structure on the vector space of cyclic words in $N$ graded letters, which extends the well-known Schedler’s necklace Lie bialgebra structure from the formality theory of the Goldman–Turaev Lie bialgebra in genus zero. Second, we introduced new (in general, nontrivial) operations in string topology. Given any closed connected and simply connected manifold $M$ of dimension $\geq 4$. We show that the reduced equivariant homology $\bar {H}_{\bullet }^{S^1}(LM)$ of the spacej $LM$ of free loops in $M$ carries a canonical representation of the dg prop ${\mathcal {H}}\textit {olieb}^\diamond _{2-n}$ on $\bar {H}_{\bullet }^{S^1}(LM)$ controlled by four ribbon hypergraphs explicitly shown in this paper.

LINEARIZATION OF POISSON–LIE STRUCTURES ON THE 2D EUCLIDEAN AND (1 + 1) POINCARÉ GROUPS

The paper deals with linearization problem of Poisson-Lie structures on the \((1+1)\) Poincaré and \(2D\) Euclidean groups. We construct the explicit form of linearizing coordinates of all these Poisson-Lie structures. For this, we calculate all Poisson-Lie structures on these two groups mentioned above, through the correspondence with Lie Bialgebra structures on their Lie algebras which we first determine.

3-dimensional \u039b-BMS symmetry and its deformations

In this paper we study quantum group deformations of the infinite dimensional symmetry algebra of asymptotically AdS spacetimes in three dimensions. Building on previous results in the finite dimensional subalgebras we classify all possible Lie bialgebra structures and for selected examples we explicitely construct the related Hopf algebras. Using cohomological arguments we show that this construction can always be performed by a so-called twist deformation. The resulting structures can be compared to the well-known κ-Poincaré Hopf algebras constructed on the finite dimensional Poincaré or (anti) de Sitter algebra. The dual κ Minkowski spacetime is supposed to describe a specific non-commutative geometry. Importantly, we find that some incarnations of the κ-Poincaré can not be extended consistently to the infinite dimensional algebras. Furthermore, certain deformations can have potential physical applications if subalgebras are considered. Since the conserved charges associated with asymptotic symmetries in 3-dimensional form a centrally extended algebra we also discuss briefly deformations of such algebras. The presence of the full symmetry algebra might have observable consequences that could be used to rule out these deformations.

Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces

Quantum homogeneous spaces are noncommutative spaces with quantum group covariance. Their semiclassical counterparts are Poisson homogeneous spaces, which are quotient manifolds of Lie groups M = G/H equipped with an additional Poisson structure π which is compatible with a Poisson–Lie structure Π on G. Since the infinitesimal version of Π defines a unique Lie bialgebra structure δ on the Lie algebra , we exploit the idea of Lie bialgebra duality in order to study the notion of complementary dual homogeneous space M ⊥ = G*/H ⊥ of a given homogeneous space M with respect to a coisotropic Lie bialgebra. Then, by considering the natural notions of reductive and symmetric homogeneous spaces, we extend these concepts to M ⊥ thus showing that an even richer duality framework between M and M ⊥ arises from them. In order to analyze physical implications of these notions, the case of M being a Minkowski or (anti-) de Sitter Poisson homogeneous spacetime is fully studied, and the corresponding complementary dual reductive and symmetric spaces M ⊥ are explicitly constructed in the case of the well-known κ-deformation, where the cosmological constant Λ is introduced as an explicit parameter in order to describe all Lorentzian spaces simultaneously. In particular, the fact that M ⊥ is a reductive space is shown to provide a natural condition for the representation theory of the quantum analogue of M that ensures the existence of physically meaningful uncertainty relations between the noncommutative spacetime coordinates. Finally, despite these dual spaces M ⊥ are not endowed in general with a G*-invariant metric, we show that their geometry can be described by making use of K-structures.

Classification of classical twists of the standard Lie bialgebra structure on a loop algebra

The standard Lie bialgebra structure on an affine Kac–Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin–Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang–Baxter equation (CYBE) with two parameters. Then, using the algebro–geometric theory of CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by Belavin and Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of CYBE.

Poisson algebras and symmetric Leibniz bialgebra structures on oscillator Lie algebras

Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures whose underlying Lie algebra is an oscillator Lie algebra. We give also all the symmetric Leibniz bialgebra structures whose underlying Lie bialgebra structure is a Lie bialgebra structure on an oscillator Lie algebra. We derive some geometric consequences on oscillator Lie groups.

Lie super-bialgebra structures on the N = 2 super-BMS3 algebra

In this paper, Lie super-bialgebra structures on the [Formula: see text] super-BMS3 algebra are investigated. By computing derivations from the super-BMS3 to the tensor product of its adjoint module, we obtain that all Lie bialgebra structures on the [Formula: see text] super-BMS3 Algebra are triangular coboundary.

Multi-oriented props and homotopy algebras with branes

We introduce a new category of differential graded multi-oriented props whose representations (called homotopy algebras with branes) in a graded vector space require a choice of a collection of k linear subspaces in that space, k being the number of extra orientations (if $$k=0$$ this structure recovers an ordinary prop); symplectic vector spaces equipped with k Lagrangian subspaces play a distinguished role in this theory. Manin triples is a classical example of an algebraic structure (concretely, a Lie bialgebra structure) given in terms of a vector space and its subspace; in the context of this paper, Manin triples are precisely symplectic Lagrangian representations of the 2-oriented generalization of the classical operad of Lie algebras. In a sense, the theory of multi-oriented props provides us with a far reaching strong homotopy generalization of Manin triples type constructions. The homotopy theory of multi-oriented props can be quite non-trivial (and different from that of ordinary props). The famous Grothendieck–Teichmuller group acts faithfully as homotopy non-trivial automorphisms on infinitely many multi-oriented props, a fact which motivated much the present work as it gives us a hint to a non-trivial deformation quantization theory in every geometric dimension $$d\ge 4$$ generalizing to higher dimensions Drinfeld–Etingof–Kazhdan’s quantizations of Lie bialgebras (the case $$d=3$$) and Kontsevich’s quantizations of Poisson structures (the case $$d=2$$).

The deformed twisted Heisenberg–Virasoro type Lie bialgebra

In this article, we investigate Lie bialgebra structures on the deformed twisted Heisenberg–Virasoro Lie algebra. Sufficient and necessary conditions for this type Lie bialgebra structures to be triangular coboundary are given.Communicated by K. C. Misra