Bialgebra Structures Research Articles

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.

Ore Extensions for the Sweedler’s Hopf Algebra H4

The aim of this paper is to classify all Hopf algebra structures on the quotient of Ore extensions H4[z;σ] of automorphism type for the Sweedler′s 4-dimensional Hopf algebra H4. Firstly, we calculate all equivalent classes of twisted homomorphisms (σ,J) for H4. Then we give the classification of all bialgebra (Hopf algebra) structures on the quotients of H4[z;σ] up to isomorphism.

A new invariant for finite dimensional Leibniz/Lie algebras

For an n-dimensional Leibniz/Lie algebra h over a field k we introduce a new invariant A(h), called the universal algebra of h, as a quotient of the polynomial algebra k[Xij|i,j=1,⋯,n] through an ideal generated by n3 polynomials. We prove that A(h) admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on h. The new object A(h) is the key tool in answering two open problems in Lie algebra theory. First, we prove that the automorphism group AutLbz(h) of h is isomorphic to the group U(G(A(h)o)) of all invertible group-like elements of the finite dual A(h)o. Secondly, for an abelian group G, we show that there exists a bijection between the set of all G-gradings on h and the set of all bialgebra homomorphisms A(h)→k[G]. Based on this, all G-gradings on h are explicitly classified and parameterized. A(h) is also used to prove that there exists a universal commutative Hopf algebra associated to any finite dimensional Leibniz algebra h.

Universal coacting Poisson Hopf algebras

We introduce the analogue of Manin’s universal coacting (bialgebra) Hopf algebra for Poisson algebras. First, for two given Poisson algebras P and U, where U is finite dimensional, we construct a Poisson algebra $$\mathcal {B}(P,\, U)$$ together with a Poisson algebra homomorphism $$\psi _{\mathcal {B}(P,\,U)} :P \rightarrow U \otimes \mathcal {B}(P,\, U)$$ satisfying a suitable universal property. $$\mathcal {B}(P,\, U)$$ is shown to admit a Poisson bialgebra structure for any pair of Poisson algebra homomorphisms subject to certain compatibility conditions. If $$P=U$$ is a finite dimensional Poisson algebra then $$\mathcal {B}(P) = \mathcal {B}(P,\, P)$$ admits a unique Poisson bialgebra structure such that $$\psi _{\mathcal {B}(P)}$$ becomes a Poisson comodule algebra and, moreover, the pair $$\bigl (\mathcal {B}(P),\, \psi _{\mathcal {B}(P)}\bigl )$$ is the universal coacting bialgebra of P. The universal coacting Poisson Hopf algebra $$\mathcal {H}(P)$$ on P is constructed as the initial object in the category of Poisson comodule algebra structures on P by using the free Poisson Hopf algebra on a Poisson bialgebra (Agore in J Math Phys 10:083502, 2014).

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

Quantization of continuum Kac–Moody algebras

Continuum Kac-Moody algebras have been recently introduced by the authors and O. Schiffmann. These are Lie algebras governed by a continuum root system, which can be realized as uncountable colimits of Borcherds-Kac-Moody algebras. In this paper, we prove that any continuum Kac-Moody algebra is canonically endowed with a non-degenerate invariant bilinear form. The positive and negative Borel subalgebras form a Manin triple with respect to this pairing, inducing on the continuum Kac-Moody algebra a topological quasi-triangular Lie bialgebra structure. We then construct an explicit quantization, which we refer to as a continuum quantum group, and we show that the latter is similarly realized as an uncountable colimit of Drinfeld-Jimbo quantum groups.

Bialgebra structures on table algebras

ABSTRACT A bialgebra is both a unital associative algebra and a coassociative coalgebra A over a field K whose algebra and coalgebra structures satisfy certain compatibility conditions. A table algebra A is an algebra over with distinguish basis and involution that satisfies certain conditions. In this paper, we establish a process to find bialgebra structures on commutative table algebras and using that we find bialgebra structures on table algebras of dimensions 2 and 3.

An algebraic study of multivariable integration and linear substitution

We set up an algebraic theory of multivariable integration, based on a hierarchy of Rota–Baxter operators and an action of the matrix monoid as linear substitutions. Given a suitable coefficient domain with a bialgebra structure, this allows us to build an operator ring that acts naturally on the given Rota–Baxter hierarchy. We conjecture that the operator relations are a noncommutative Gröbner–Shirshov basis for the ideal they generate.

Double Constructions of Compatible Associative Algebras

A compatible associative algebra is a pair of associative algebras satisfying that any linear combination of the two associative products is still an associative product. We construct a compatible associative algebra with a decomposition into the direct sum of the underlying vector space of another compatible associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant. This compatible associative algebra is equivalent to a certain bialgebra structure of compatible associative algebras, which is an analogue of a Lie bialgebra. Many properties of the bialgebra are presented. In particular, the coboundary bialgebra theory leads to the system of associative Yang–Baxter equations in compatible associative algebras, which is an analogue of the classical Yang–Baxter equation in a Lie algebra. Furthermore, the bialgebra can also be regarded as a “compatible version” of antisymmetric infinitesimal bialgebras, that is, a pair of antisymmetric infinitesimal bialgebras satisfying any linear combination of them is still an antisymmetric infinitesimal bialgebra.

Dual Lie Bialgebra Structures of W-algebra W(2, 2) Type

In the present paper, we investigate the dual Lie coalgebras of the centerless W(2, 2) algebra by studying the maximal good subspaces. Based on this, we construct the dual Lie bialgebra structures of the centerless W(2, 2) Lie bialgebra. As by-products, four new infinite dimensional Lie algebras are obtained.

Drinfel’d double structures for Poincaré and Euclidean groups

All non-isomorphic three-dimensional Poisson homogeneous Euclidean spaces are constructed and analyzed, based on the classification of coboundary Lie bialgebra structures of the Euclidean group in 3-dimensions, and the only Drinfel’d double structure for this group is explicitly given. The similar construction for the Poincaré case is reviewed and the striking differences between the Lorentzian and Euclidean cases are underlined. Finally, the contraction scheme starting from Drinfel’d double structures of the Lie algebra is presented.

Lie bialgebra structures on generalized loop Schrödinger-Virasoro algebras

We give a classification of Lie bialgebra structures on generalized loop Schrodinger-Virasoro algebras $$\mathfrak{sv}$$ . Then we find out that not all Lie bialgebra structures on generalized loop Schrodinger-Virasoro algebras $$\mathfrak{sv}$$ are triangular coboundary.

A new branch of bialgebraic structures: UP-bialgebras

The notion of bialgebraic structures was discussed by Vasantha Kandasamy [Bialgebraic structures and Smarandache bialgebraic structures. India: American Research Press; 2003]. The main target of this paper is to introduce the notions of a KU-bialgebra, a KP-bialgebra, a PK-bialgebra, and a UP-bialgebra and the notions of a UP-bisubalgebra, a UP-bifilter, a UP-biideal, and a strongly UP-biideal of UP-bialgebras, and prove the generalization of the notions and some results related to a UP-subalgebra, a UP-filter, a UP-ideal, and a strongly UP-ideal of UP-algebras. Furthermore, we introduce the notion of a UP-bihomomorphism and study the image and inverse image of a UP-bisubalgebra, a UP-bifilter, a UP-biideal, and a strongly UP-biideal of UP-bialgebras under a UP-bihomomorphism. Finally, we have the generalization diagram of KU/KP/PK/UP-bialgebras (see Figure 1) and the diagram of special subsets of UP-bialgebras (see Figure 2).

On Ideals of Bi-Algebras

In this paper, we discuss normal subalgebras in BI-algebras and obtain the quotient BI-algebra which is useful for the study of structures of BI-algebras. Moreover, we obtain several conditions for obtaining BI-algebras on the non-negative real numbers by using an analytic methods.

Braided Yangians

Yangian-like algebras associated with current R-matrices different from the Yang ones are introduced. These algebras are of two types. The so-called braided Yangians are close to the Reflection Equation algebras, arising from involutive or Hecke symmetries. The Yangians of RTT type are close to the corresponding RTT algebras. Some properties of these two classes of the Yangian-like algebras are studied. Thus, evaluation morphisms for them are constructed, their bi-algebra structures are described, and quantum analogs of certain symmetric polynomials, in particular, the quantum determinant, are introduced. It is proved that in any braided Yangian this determinant is always central, whereas, in general, this is not true for the Yangians of RTT type. Analogs of the Cayley–Hamilton–Newton identities in the braided Yangians are exhibited. A bosonic realization of the braided Yangians is performed.

FREE MODIFIED ROTA-BAXTER ALGEBRAS AND HOPF ALGEBRAS

The notion of a modified Rota-Baxter algebra comes from the combination of those of a Rota-Baxter algebra and a modified Yang-Baxter equation. In this paper, we first construct free modified Rota-Baxter algebras. We then equip a free modified Rota-Baxter algebra with a bialgebra structure by a cocycle construction. Under the assumption that the generating algebra is a connected bialgebra, we further equip the free modified Rota-Baxter algebra with a Hopf algebra structure.

Weighted Khovanov-Lauda-Rouquier Algebras

In this paper, we define a generalization of Khovanov–Lauda–Rouquier algebras which we call weighted Khovanov–Lauda–Rouquier algebras . We show that these algebras carry many of the same structures as the original Khovanov–Lauda–Rouquier algebras, including induction and restriction functors which induce a twisted bialgebra structure on their Grothendieck groups. We also define natural steadied quotients of these algebras, which in an important special cases give categorical actions of an associated Lie algebra. These include the algebras categorifying tensor products and Fock spaces defined by the author and C. Stroppel [ B. Webster , Mem. Am. Math. Soc. 1191, iii-vi, 146 p. (2017; Zbl 07000045), p. 141, and C. Stroppel and B. Webster , “Quiver Schur algebras and q -Fock space”, Preprint, ]. For symmetric Cartan matrices, weighted KLR algebras also have a natural geometric interpretation as convolution algebras, generalizing that for the original KLR algebras by M. Varagnolo and E. Vasserot [J. Reine Angew. Math. 659, 67–100 (2011; Zbl 1229.17019)]; this result has positivity consequences important in the theory of crystal bases. In this case, we can also relate the Grothendieck group and its bialgebra structure to the Hall algebra of the associated quiver.

On the Classification of Lie Bialgebras by Cohomological Means

We approach the classification of Lie bialgebra structures on simple Lie algebras from the viewpoint of descent and non-abelian cohomology. We achieve a description of the problem in terms of faithfully flat cohomology over an arbitrary ring over \mathbb{Q} , and solve it for Drinfeld-Jimbo Lie bialgebras over fields of characteristic zero. We consider the classification up to isomorphism, as opposed to equivalence, and treat split and non-split Lie algebras alike. We moreover give a new interpretation of scalar multiples of Lie bialgebras hitherto studied using twisted Belavin-Drinfeld cohomology.