Bialgebra Structures Research Articles

Universal coacting Hopf algebra of a finite dimensional Lie-Yamaguti algebra

M. E. Sweedler first constructed a universal Hopf algebra of an algebra. It is known that the dual notions to the existing ones play a dominant role in Hopf algebra theory. Yu. I. Manin and D. Tambara introduced the dual notion of Sweedler's construction in separate works. In this paper, we construct a universal algebra for a finite-dimensional Lie-Yamaguti algebra. We demonstrate that this universal algebra possesses a bialgebra structure, leading to a universal coacting Hopf algebra for a finite-dimensional Lie-Yamaguti algebra. Additionally, we develop a representation-theoretic version of our results. As an application, we characterize the automorphism group and classify all abelian group gradings of a finite-dimensional Lie-Yamaguti algebra.

Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras

It is known that the operads of perm algebras and pre-Lie algebras are the Koszul dual each other and hence there is a Lie algebra structure on the tensor product of a perm algebra and a pre-Lie algebra. Conversely, we construct a special perm algebra structure and a special pre-Lie algebra structure on the vector space of Laurent polynomials such that the tensor product with a pre-Lie algebra and a perm algebra being a Lie algebra structure characterizes the pre-Lie algebra and the perm algebra respectively. This is called the affinization of a pre-Lie algebra and a perm algebra respectively. Furthermore we extend such correspondences to the context of bialgebras, that is, there is a bialgebra structure for a perm algebra or a pre-Lie algebra which could be characterized by the fact that its affinization by a quadratic pre-Lie algebra or a quadratic perm algebra respectively gives an infinite-dimensional Lie bialgebra. In the case of perm algebras, the corresponding bialgebra structure is called a perm bialgebra, which can be independently characterized by a Manin triple of perm algebras as well as a matched pair of perm algebras. The notion of the perm Yang-Baxter equation is introduced, whose symmetric solutions give rise to perm bialgebras. There is a correspondence between symmetric solutions of the perm Yang-Baxter equation in perm algebras and certain skew-symmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras induced from the perm algebras. In the case of pre-Lie algebras, the corresponding bialgebra structure is a pre-Lie bialgebra which is well-constructed. The similar correspondences for the related structures are given.

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.

Everybody knows what a normal gabi-algebra is

Let A be a k-algebra over a commutative ring k. By the renowned Tannaka-Kreĭn reconstruction, liftings of the monoidal structure from Mk to MA correspond to bialgebra structures on A and liftings of the closed monoidal structure correspond to Hopf algebra structures on A. In this paper, we determine conditions on A that correspond to liftings of the closed structure alone, i.e. without considering the monoidal one, which lead to the notion of what we call a gabi-algebra. First, we tackle the question from the general perspective of monads, then we focus on the set-theoretic and the linear setting. Our main and most surprising result is that a normal gabi-algebra, that is an algebra A whose category of modules is (associative and unital normal) closed with closed forgetful functor to Mk, is automatically a Hopf algebra (thus justifying our title).

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.

Orthogonal Polynomial Duality and Unitary Symmetries of Multi-species ASEP$$(q,\varvec{\theta })$$ and Higher-Spin Vertex Models via $$^*$$-Bialgebra Structure of Higher Rank Quantum Groups

Orthogonal Polynomial Duality and Unitary Symmetries of Multi-species ASEP$$(q,\varvec{\theta })$$ and Higher-Spin Vertex Models via $$^*$$-Bialgebra Structure of Higher Rank Quantum Groups

Eight Times Four Bialgebras of Hypergraphs, Cointeractions, and Chromatic Polynomials

Abstract The bialgebra of hypergraphs, a generalization of W. Schmitt’s Hopf algebra of graphs [25], is shown to have a cointeracting bialgebra structure, giving a double bialgebra in the sense of L. Foissy, who has recently proven [15] that there is then a unique double bialgebra morphism to the double bialgebra structure on the polynomial ring ${\mathbb Q}[x]$. We show that the associated polynomial is the hypergraph chromatic polynomial. Moreover, hypergraphs occur in quartets: There is a dual, a complement, and a dual complement hypergraph. These correspondences are involutions and give rise to three other double bialgebras, and three more chromatic polynomials. In addition to these two quartets of bialgebras we give six more, including recent bialgebras of hypergraphs introduced by M. Aguiar and F. Ardila [1] and by L. Foissy [17].

Quasi-triangular pre-Lie bialgebras, factorizable pre-Lie bialgebras and Rota-Baxter pre-Lie algebras

In this paper, first we introduce the notions of quasi-triangular pre-Lie bialgebras and factorizable pre-Lie bialgebras. A factorizable pre-Lie bialgebra leads to a factorization of the underlying pre-Lie algebra. We show that the symplectic double of a pre-Lie bialgebra naturally enjoys a factorizable pre-Lie bialgebra structure. Then we give the Rota-Baxter characterization of factorizable pre-Lie bialgebras. More precisely, we introduce the notion of quadratic Rota-Baxter pre-Lie algebras and show that there is a one-to-one correspondence between factorizable pre-Lie bialgebras and quadratic Rota-Baxter pre-Lie algebras. Finally, we develop the theories of matched pairs, bialgebras and Manin triples of Rota-Baxter pre-Lie algebras. In particular, a factorizable pre-Lie bialgebra gives rise to a Rota-Baxter pre-Lie bialgebra, and conversely a Rota-Baxter pre-Lie bialgebra gives rise to a factorizable pre-Lie bialgebra structure on the double space.

Relative Poisson bialgebras and Frobenius Jacobi algebras

Jacobi algebras, as the algebraic counterparts of Jacobi manifolds, are exactly the unital relative Poisson algebras. The direct approach of constructing Frobenius Jacobi algebras in terms of Manin triples is not available due to the existence of the units, and hence alternatively we replace it by studying Manin triples of relative Poisson algebras. Such structures are equivalent to certain bialgebra structures, namely, relative Poisson bialgebras. The study of coboundary cases leads to the introduction of the relative Poisson Yang–Baxter equation (RPYBE). Antisymmetric solutions of the RPYBE give coboundary relative Poisson bialgebras. The notions of \mathcal{O} -operators of relative Poisson algebras and relative pre-Poisson algebras are introduced to give antisymmetric solutions of the RPYBE. A direct application is that relative Poisson bialgebras can be used to construct Frobenius Jacobi algebras, and in particular, there is a construction of Frobenius Jacobi algebras from relative pre-Poisson algebras.

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.

Loop coproduct in Morse and Floer homology

By a well-known theorem of Viterbo, the symplectic homology of the cotangent bundle of a closed manifold is isomorphic to the homology of its loop space. In this paper, we extend the scope of this isomorphism in several directions. First, we give a direct definition of Rabinowitz loop homology in terms of Morse theory on the loop space and prove that its product agrees with the pair-of-pants product on Rabinowitz Floer homology. The proof uses compactified moduli spaces of punctured annuli. Second, we prove that, when restricted to positive Floer homology, resp. loop space homology relative to the constant loops, the Viterbo isomorphism intertwines various constructions of secondary pair-of-pants coproducts with the loop homology coproduct. Third, we introduce reduced loop homology, which is a common domain of definition for a canonical reduction of the loop product and for extensions of the loop homology coproduct which together define the structure of a commutative cocommutative unital infinitesimal anti-symmetric bialgebra. Along the way, we show that the Abbondandolo–Schwarz quasi-isomorphism going from the Floer complex of quadratic Hamiltonians to the Morse complex of the energy functional can be turned into a filtered chain isomorphism using linear Hamiltonians and the square root of the energy functional.

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.

Quasi-triangular and factorizable antisymmetric infinitesimal bialgebras

In this paper, we introduce the notions of quasi-triangular and factorizable antisymmetric infinitesimal bialgebras. A factorizable antisymmetric infinitesimal bialgebra leads to a factorization of the underlying associative algebra. We show that the associative double of an antisymmetric infinitesimal bialgebra naturally enjoys a factorizable antisymmetric infinitesimal bialgebra structure. Then we introduce the notion of symmetric Rota-Baxter Frobenius algebras, and show that there is a one-to-one correspondence between factorizable antisymmetric infinitesimal bialgebras and symmetric Rota-Baxter Frobenius algebras. Finally, we show that a symmetric Rota-Baxter Frobenius algebra can give rise to an isomorphism from the regular representation to the coregular representation of a Rota-Baxter associative algebra.

Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras

We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double constructions of differential Frobenius algebras as well as matched pairs of differential algebras. Antisymmetric solutions of an analogue of associative Yang-Baxter equation in differential algebras provide differential ASI bialgebras, whereas in turn the notions of O-operators of differential algebras and differential dendriform algebras are also introduced to produce the former. On the other hand, the notion of a coherent derivation on an ASI bialgebra is introduced as an equivalent structure of a differential ASI bialgebra. They include derivations on ASI bialgebras and the set of coherent derivations on an ASI bialgebra composes a Lie algebra which is the Lie algebra of the Lie group consisting of coherent automorphisms on this ASI bialgebra. Finally, we apply the study of differential ASI bialgebras to Poisson bialgebras, extending the construction of Poisson algebras from commutative differential algebras with two commuting derivations to the context of bialgebras, which is consistent with the well constructed theory of Poisson bialgebras. In particular, we construct Poisson bialgebras from differential Zinbiel algebras.

Twisting of properads

We study T. Willwacher's twisting endofunctor tw in the category of dg prop(erad)s P under the operad of (strongly homotopy) Lie algebras, i:Lie→P. It is proven that if P is a properad under properad of Lie bialgebras Lieb, then the associated twisted properad twP becomes in general a properad under quasi-Lie bialgebras (rather than under Lieb). This result implies that the cyclic cohomology of any cyclic homotopy associative algebra has in general an induced structure of a quasi-Lie bialgebra. We show that the cohomology of the twisted properad twLieb is highly non-trivial — it contains the cohomology of the so called hairy graph complex introduced and studied recently in the context of the theory of long knots and the theory of moduli spaces Mg,n of algebraic curves of arbitrary genus g with n punctures.Using a polydifferential functor from the category of props to the category of operads, we introduce and study two new twisting endofunctors, one in the category dg prop(erad)s P under the minimal resolution of Lieb, and one for the involutive version of Lieb. We compute the cohomology of the associated deformation complexes, and discuss their applications in string topology.

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.

Quasi-Lie bialgebras of loops in quasisurfaces

We discuss natural operations on loops in a quasisurface and show that these operations define a structure of a quasi-Lie bialgebra in the module generated by the set of free homotopy classes of noncontractible loops.