Bialgebra Structures Research Articles

Lie bialgebras of a family of Lie algebras of Block type

Lie bialgebra structures on a family of Lie algebras of Block type are shown to be triangular coboundary.

The Classical r-Matrix of AdS/CFT and its Lie Bialgebra Structure

In this paper we investigate the algebraic structure of AdS/CFT in the strong-coupling limit. We propose an expression for the classical r-matrix with (deformed) \({\mathfrak{u}(2|2)}\) symmetry, which leads to a quasi-triangular Lie bialgebra as the underlying symmetry algebra. On the fundamental representation our r-matrix coincides with the classical limit of the quantum R-matrix.

On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization

We study classical twists of Lie bialgebra structures on the polynomial current algebra \({\mathfrak{g}[u]}\), where \({\mathfrak{g}}\) is a simple complex finite-dimensional Lie algebra. We focus on the structures induced by the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. It turns out that quasi-trigonometric r-matrices fall into classes labelled by the vertices of the extended Dynkin diagram of \({\mathfrak{g}}\). We give the complete classification of quasi-trigonometric r-matrices belonging to multiplicity free simple roots (which have coefficient 1 in the decomposition of the maximal root). We quantize solutions corresponding to the first root of \({\mathfrak{sl}(n)}\).

Lie Bialgebra Structures on Lie Algebras of Generalized Weyl Type

Lie bialgebra structures on Lie algebras of generalized Weyl type are studied. They are shown to be triangular coboundary.

ON LIE BIALGEBRAS OF LOOPS ON ORIENTABLE SURFACES

Goldman [2] and Turaev [4] found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of loops on an orientable surface. Chas [1] by the aid of the computer, found a negative answer to Turaev's question about the characterization of the classes with cobracket zero as multiples of simple classes, in every surface of negative Euler characteristic and positive genus. However, she left open Turaev's conjecture, namely if, for genus zero, every class with cobracket zero is a multiple of a simple class. The aim of this paper is to give a positive answer to this conjecture.

Deformations of schemes and other bialgebraic structures

There has long been a philosophy that every deformation problem in characteristic zero should be governed by a differential graded Lie algebra (DGLA). In this paper, we show how to construct a Simplicial Deformation Complex (SDC) governing any bialgebraic deformation problem. Examples of such problems are deformations of a Hopf algebra, or of an arbitrary scheme. In characteristic zero, SDCs and DGLAs are shown to be equivalent.

Strongly Homotopy Lie Bialgebras and Lie Quasi-bialgebras

Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer–Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of Kosmann–Schwarzbach. This approach provides a definition of an L ∞-(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L ∞-algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L ∞ (quasi)bialgebra structure on V as a differential operator Q on V, self-commuting with respect to the big bracket. Finally, we establish an L ∞-version of a Manin (quasi) triple and get a correspondence theorem with L ∞-(quasi)bialgebras.

Hamiltonian type Lie bialgebras

We first prove that, for any generalized Hamiltonian type Lie algebra $$\mathcal{H}$$ , the first cohomology group $$H^1 (\mathcal{H},\mathcal{H} \otimes \mathcal{H})$$ is trivial. We then show that all Lie bialgebra structures on $$\mathcal{H}$$ are triangular

Classicalrmatrix of thesu(2|2)super Yang-Mills spin chain

In this note we straightforwardly derive and make use of the quantum $R$ matrix for the $\mathfrak{s}\mathfrak{u}(2|2)$ super Yang-Mills spin chain in the manifest $\mathfrak{s}\mathfrak{u}(1|2)$-invariant formulation, which solves the standard quantum Yang-Baxter equation, in order to obtain the correspondent (undressed) classical $r$ matrix from the first order expansion in the ``deformation'' parameter $2\ensuremath{\pi}/\sqrt{\ensuremath{\lambda}}$ and check that this last solves the standard classical Yang-Baxter equation. We analyze its bialgebra structure, its dependence on the spectral parameters, and its pole structure. We notice that it still preserves an $\mathfrak{s}\mathfrak{u}(1|2)$ subalgebra, thereby admitting an expression in terms of a combination of projectors, which spans only a subspace of $\mathfrak{s}\mathfrak{u}(1|2)\ensuremath{\bigotimes}\mathfrak{s}\mathfrak{u}(1|2)$. We study the residue at its simple pole at the origin and comment on the applicability of the classical Belavin-Drinfeld type of analysis.

Lie Bialgebras of Generalized WITT Type, II

In an article 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 a recent article by Song and Su, Lie bialgebra structures on graded Lie algebras of generalized Witt type with finite dimensional homogeneous components were considered. In this article we consider Lie bialgebra structures on the graded Lie algebras of generalized Witt type with infinite dimensional homogeneous components. By proving that the first cohomology group H1(𝒲, 𝒲 ⊗ 𝒲) is trivial for any graded Lie algebras 𝒲 of generalized Witt type with infinite dimensional homogeneous components, we obtain that all such Lie bialgebras are triangular coboundary.

Small bialgebras with a projection

Let A be a bialgebra with an H-bilinear coalgebra projection over an arbitrary subbialgebra H with antipode. In characteristic zero, we completely describe the bialgebra structure of A whenever H is either f.d. or cosemisimple and the H-coinvariant part R of A is connected with one-dimensional space of primitive elements.

Quantizations of generalized-Witt algebra and of Jacobson–Witt algebra in the modular case

We quantize the generalized-Witt algebra in characteristic 0 with its Lie bialgebra structures discovered by Song–Su [G. Song, Y. Su, Lie bialgebras of generalized-Witt type, arXiv: math.QA/0504168, Sci. China Ser. A 49 (4) (2006) 533–544]. Via a modulo p reduction and a modulo “ p-restrictedness” reduction process, we get 2 n − 1 families of truncated p-polynomial noncocommutative deformations of the restricted universal enveloping algebra of the Jacobson–Witt algebra W ( n ; 1 ̲ ) (for the Cartan type simple modular restricted Lie algebra of W type). They are new families of noncommutative and noncocommutative Hopf algebras of dimension p 1 + n p n in characteristic p. Our results generalize a work of Grunspan [C. Grunspan, Quantizations of the Witt algebra and of simple Lie algebras in characteristic p, J. Algebra 280 (2004) 145–161] in rank n = 1 case in characteristic 0. In the modular case, the argument for a refined version follows from the modular reduction approach (different from [C. Grunspan, Quantizations of the Witt algebra and of simple Lie algebras in characteristic p, J. Algebra 280 (2004) 145–161]) with some techniques from the modular Lie algebra theory.

Categorical Methods in Hopf Algebras

The main idea behind many notions in Ring Theory and other disciplines can be formulated in a natural way in a categorical setting. For example, a natural way to introduce the notion of bialgebra is the following: bialgebra structures on an algebra A correspond bijectively to monoidal structures on the category of representations of A, such that the forgetful functor to the category of vector spaces is strongly monoidal. If we require, in addition, that the category of representations has duality, then we recover the definition of a Hopf algebra. Other concepts, that appeared more recently in connection to quantum group theory, such as quasi-triangular Hopf algebras, quasi-Hopf algebras, weak Hopf algebras and Hopf algebroids, can be explained in a similar way, using (braided) monoidal categories. This special issue contains a collection of recent articles on Hopf algebras and related topics, reflecting this philosophy. An important recent trend is the revival of the theory of corings. Corings can be considered as coalgebras over non-commutative rings; they were introduced by Sweedler in 1975. A renewed interest in the topic began in 2000, with a paper by Brzezinski, in which it was explained how a series of results on Hopf modules and related topics can be unified and reformulated more elegantly in terms of corings. Six contributions in this issue are directly connected to corings. The most categorical of them is the paper by Gomez–Torrecillas, explaining connections between Galois theory for corings and comonads. Three other papers (by Bohm and Brzezinski, by Kadison, and by Panaite and Van Oystaeyen) are about Hopf algebroids. Hopf algebroids can be viewed as Hopf algebras over non-commutative rings, but, as indicated above, they can be interpreted in terms of monoidal categories.

The 4th structure

In this paper we describe all Lie bialgebra structures on the polynomial Lie algebra $\mathbf{g}[u]$, where $\mathbf{g}$ is a simple, finite dimensional, complex Lie algebra. The results are based on an unpublished paper Montaner and Zelmanov. Further, we introduce quasi-rational solutions of the CYBE and describe all quasi-rational $r$-matrices for $\mathbf{sl}(2)$.

Renormalization as a functor on bialgebras

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T ( T ( B ) + ) , the double tensor algebra of B , with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S ( S ( B ) + ) , the double symmetric algebra of B , into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of S 1 ( B ) are not renormalized, i.e., when Feynman diagrams containing one single vertex are not renormalized. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S ( S ( B ) + ) and the Faà di Bruno bialgebra of composition of series. The relation with the Connes–Moscovici Hopf algebra is given. Finally, the bialgebra S ( S ( B ) + ) is shown to give the same results as the standard renormalization procedure for the scalar field.

The Drinfeld doublegl(n) ⊕tn

The two isomorphic Borel subalgebras of gl(n), realized on upper and lower triangular matrices, allow us to consider the gl(n) \opus t_n algebra as a self-dual Drinfeld double. Compatibility conditions impose the choice of an orthonormal basis in the Cartan subalgebra and fix the basis of gl(n). A natural Lie bialgebra structure on gl(n) is obtained, that offers a new perspective for its standard quantum deformation.

Lie Bialgebras of Generalized Virasoro–like Type

In this paper, Lie bialgebra structures on generalized Virasoro–like algebras are studied. It is proved that all such Lie bialgebras are triangular coboundary.

Lie bialgebras of generalized Witt type

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.

Classical quasi-trigonometric r−matrices of Cremmer-Gervais type and their quantization

We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang-Baxter equation. The method is based on so-called affinization of certain seaweed algebras and their quantum analogues.

Classical quasi-trigonometric r-matrices of Cremmer-Gervais type and their quantization

We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang-Baxter equation. The method is based on so-called affinization of certain seaweed algebras and their quantum analogues.