Manin Triples Research Articles

Equivalent Notions in the context of compatible Endo-Lie Algebras

In this article, we introduce a notion of compatibility between two Endo-Lie algebras defined on the same linear space. Compatibility means that any linear combination of the two structures always induces a new Endo-Lie algebras structure. In this case of compatibility, we show that the notions of bialgebras, standard Manin triples and matched pairs are equivalent. We find this equivalence for the case of compatible Lie algebras since this is a particular case of compatible Endo-Lie algebras.

Manin triples associated to n-Lie bialgebras

In this paper, we study the Manin triples associated to [Formula: see text]-Lie bialgebras. We introduce the concept of operad matrices for [Formula: see text]-Lie bialgebras. In particular, by studying a special case of operad matrices, it leads to the notion of local cocycle [Formula: see text]-Lie bialgebras. Furthermore, we establish a one-to-one correspondence between the double of [Formula: see text]-Lie bialgebras and Manin triples of [Formula: see text]-Lie algebras.

Bialgebras via Manin triples of compatible mock-Lie algebras

Motivated by recent work on compatible mock-Lie algebras such that any linear combination of the two mock-Lie structures is a mock-Lie structure, in this paper we introduce compatible mock-Lie bialgebras as an analogue of compatible Lie bialgebras. They can also be regarded as a “compatible version” of mock-Lie bialgebras. Constructions of compatible mock-Lie bialgebras are presented via Manin triples and matched pairs.

Symplectic structures, product structures and complex structures on Leibniz algebras

In this paper, a symplectic structure on a Leibniz algebra is defined to be a symmetric nondegenerate bilinear form satisfying certain compatibility condition, and a phase space of a Leibniz algebra is defined to be a symplectic Leibniz algebra satisfying certain conditions. We show that a Leibniz algebra has a phase space if and only if there is a compatible Leibniz-dendriform algebra, and phase spaces of Leibniz algebras are one-to-one corresponds to Manin triples of Leibniz-dendriform algebras. Product (paracomplex) structures and complex structures on Leibniz algebras are studied in terms of decompositions of Leibniz algebras. A para-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a paracomplex structure satisfying a compatibility condition. We show that a symplectic Leibniz algebra admits a para-Kähler structure if and only if the Leibniz algebra is the direct sum of two Lagrangian subalgebras as vector spaces. A complex product structure on a Leibniz algebra consists of a complex structure and a product structure satisfying a compatibility condition. A pseudo-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a complex structure satisfying a compatibility condition. Various properties and relations of complex product structures and pseudo-Kähler structures are studied. In particular, Leibniz-dendriform algebras give rise to complex product structures and pseudo-Kähler structures naturally.

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.

Manin Triples and Bialgebras of Left-Alia Algebras Associated with Invariant Theory

A left-Alia algebra is a vector space together with a bilinear map satisfying the symmetric Jacobi identity. Motivated by invariant theory, we first construct a class of left-Alia algebras induced by twisted derivations. Then, we introduce the notions of Manin triples and bialgebras of left-Alia algebras. Via specific matched pairs of left-Alia algebras, we figure out the equivalence between Manin triples and bialgebras of left-Alia algebras.

Rota–Baxter mock-Lie bialgebras and related structures

The aim of this paper is to introduce the notion of Rota–Baxter mock-Lie bialgebras [Formula: see text] and their admissibility conditions in terms of dual representations [Formula: see text]. Next, we show that Rota–Baxter mock-Lie bialgebras are characterized by matched pairs and Manin triples of Rota–Baxter mock-Lie algebras. Furthermore, the coboundary case leads to the introduction of the admissible mock-Lie Yang–Baxter equation in Rota–Baxter mock-Lie algebras, whose skew-symmetric solutions are used to construct Rota–Baxter mock-Lie bialgebras.

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.

Bialgebras, the Yang–Baxter equation and Manin triples for mock-Lie algebras

The aim of this paper is to introduce the notion of a mock-Lie bialgebra which is equivalent to a Manin triple of mock-Lie algebras. The study of a special case called coboundary mock-Lie bialgebra leads to introducing the mock-Lie Yang–Baxter equation on a mock-Lie algebra which is an analogue of the classical Yang–Baxter equation on a Lie algebra. Note that a skew-symmetric solution of mock-Lie Yang–Baxter equation gives a mock-Lie bialgebra. Finally, O-operators are studied to construct a skew-symmetric solution of a mock-Lie Yang–Baxter equation.

A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie Poisson bialgebras

The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang–Baxter equation and [Formula: see text]-operators are studied.

Courant-Nijenhuis algebroids

We introduce Courant 1-derivations, which describe a compatibility between Courant algebroids and linear (1,1)-tensor fields and lead to the notion of Courant-Nijenhuis algebroids. We provide examples of Courant 1-derivations on exact Courant algebroids and show that holomorphic Courant algebroids can be viewed as special types of Courant-Nijenhuis algebroids. By considering Dirac structures, one recovers the Dirac-Nijenhuis structures of [5] (in the special case of the standard Courant algebroid) and obtains an equivalent description of Lie-Nijenhuis bialgebroids [9] via Manin triples.

Higher current algebras, homotopy Manin triples, and a rectilinear adelic complex

The notion of a Manin triple of Lie algebras admits a generalization, to dg Lie algebras, in which various properties are required to hold only up to homotopy.This paper introduces two classes of examples of such homotopy Manin triples. These examples are associated to analogs in complex dimension two of, respectively, the punctured formal 1-disc, and the complex plane with multiple punctures. The dg Lie algebras which appear include certain higher current algebras in the sense of Faonte, Hennion and Kapranov [18].We work in a ringed space we call rectilinear space, and one of the tools we introduce is a model of the derived sections of its structure sheaf, whose construction is in the spirit of the adelic complexes for schemes due to Parshin and Beilinson.

Hom-Leibniz bialgebras and BiHom-Leibniz dendriform algebras

A notion of a Hom-Leibniz bialgebra is introduced and it is shown that matched pairs of Hom-Leibniz algebras, Manin triples of Hom-Leibniz algebras and Hom-Leibniz bialgebras are equivalent in a certain sense. A notion of Hom-Leibniz dendriform algebra is established, their bimodules and matched pairs are defined and their properties and theorems about their interplay and construction are obtained. Furthermore, the concept of BiHom-Leibniz dendriform algebras is introduced and discussed, their bimodules and matched pairs are constructed and properties are described. Finally, the connections between all these algebraic structures using {mathcal {O}}-operators are shown.

Manin triples and non-degenerate anti-symmetric bilinear forms on Lie superalgebras in characteristic 2

In this paper, we introduce and develop the notion of a Manin triple for a Lie superalgebra g defined over a field of characteristic p=2. We find cohomological necessary conditions for the pair (g,g⁎) to form a Manin triple. We introduce the concept of Lie bi-superalgebras for p=2 and establish a link between Manin triples and Lie bi-superalgebras. In particular, we study Manin triples defined by a classical r-matrix with an extra condition (called an admissible classical r-matrix). A particular case is examined where g has an even invariant non-degenerate bilinear form. In this case, admissible r-matrices can be obtained inductively through the process of double extensions. In addition, we introduce the notion of double extensions of Manin triples, and show how to get a new Manin triple from an existing one.

Factorizable Lie Bialgebras, Quadratic Rota–Baxter Lie Algebras and Rota–Baxter Lie Bialgebras

In this paper, first we introduce the notion of quadratic Rota–Baxter Lie algebras of arbitrary weight, and show that there is a one-to-one correspondence between factorizable Lie bialgebras and quadratic Rota–Baxter Lie algebras of nonzero weight. Then we introduce the notions of matched pairs, bialgebras and Manin triples of Rota–Baxter Lie algebras of arbitrary weight, and show that Rota–Baxter Lie bialgebras, Manin triples of Rota–Baxter Lie algebras and certain matched pairs of Rota–Baxter Lie algebras are equivalent. The coadjoint representations and quadratic Rota–Baxter Lie algebras play important roles in the whole study. Finally we generalize some results to the Lie group context. In particular, we show that there is a one-to-one correspondence between factorizable Poisson Lie groups and quadratic Rota–Baxter Lie groups.

3-Hom–Lie Yang–Baxter Equation and 3-Hom–Lie Bialgebras

In this paper, we first introduce the notion of a 3-Hom–Lie bialgebra and give an equivalent description of the 3-Hom–Lie bialgebras, the matched pairs and the Manin triples of 3-Hom–Lie algebras. In addition, we define O-operators of 3-Hom–Lie algebras and construct solutions of the 3-Hom–Lie Yang–Baxter equation in terms of O-operators and 3-Hom–pre-Lie algebras. Finally, we show that a 3-Hom–Lie algebra has a phase space if and only if it is sub-adjacent to a 3-Hom–pre-Lie algebra.

The Classical Hom–Leibniz Yang–Baxter Equation and Hom–Leibniz Bialgebras

In this paper, we first introduce the notion of Hom–Leibniz bialgebras, which is equivalent to matched pairs of Hom–Leibniz algebras and Manin triples of Hom–Leibniz algebras. Additionally, we extend the notion of relative Rota–Baxter operators to Hom–Leibniz algebras and prove that there is a Hom–pre-Leibniz algebra structure on Hom–Leibniz algebras that have a relative Rota–Baxter operator. Finally, we study the classical Hom–Leibniz Yang–Baxter equation on Hom–Leibniz algebras and present its connection with the relative Rota–Baxter operator.

Coherent categorical structures for Lie bialgebras, Manin triples, classical r-matrices and pre-Lie algebras

Abstract The broadly applied notions of Lie bialgebras, Manin triples, classical r-matrices and 𝒪 {\mathcal{O}} -operators of Lie algebras owe their importance to the close relationships among them. Yet these notions and their correspondences are mostly understood as classes of objects and maps among the classes. To gain categorical insight, this paper introduces, for each of the classes, a notion of homomorphisms, uniformly called coherent homomorphisms, so that the classes of objects become categories and the maps among the classes become functors or category equivalences. For this purpose, we start with the notion of an endo Lie algebra, consisting of a Lie algebra equipped with a Lie algebra endomorphism. We then generalize the above classical notions for Lie algebras to endo Lie algebras. As a result, we obtain the notion of coherent endomorphisms for each of the classes, which then generalizes to the notion of coherent homomorphisms by a polarization process. The coherent homomorphisms are compatible with the correspondences among the various constructions, as well as with the category of pre-Lie algebras.

Relative Rota-Baxter operators and symplectic structures on Lie-Yamaguti algebras

In this paper, first we show that the invariant bilinear form in a quadratic Lie-Yamaguti algebra induces an isomorphism between the adjoint representation and the coadjoint representation. Then we introduce the notions of relative Rota-Baxter operators on Lie-Yamaguti algebras and pre-Lie-Yamaguti algebras. We prove that a pre-Lie-Yamaguti algebra gives rise to a Lie-Yamaguti algebra naturally and a relative Rota-Baxter operator induces a pre-Lie-Yamaguti algebra. Finally, we study symplectic structures on Lie-Yamaguti algebra, which give rise to relative Rota-Baxter operators as well as pre-Lie-Yamaguti algebras. As applications, we study phase spaces of Lie-Yamaguti algebras, and show that there is a one-to-one correspondence between phase spaces of Lie-Yamaguti algebras and Manin triples of pre-Lie-Yamaguti algebras.

A New Approach to Jordan D-Bialgebras via Jordan–Manin Triples

Jordan D-bialgebras were introduced by Zhelyabin. In this paper, we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan–Manin triples called double constructions of pseudo-euclidean Jordan algebras. We also show that a Jordan D-bialgebra leads to the Jordan Yang–Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang–Baxter equation corresponds to an antisymmetric bilinear form, which we call a Jordan symplectic form on Jordan algebras. Furthermore, there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form.