Searcharxiv⌕ Search

arXiv subjects

Yunhe Sheng

Publications and source records attributed to Yunhe Sheng.

At least 55 records · Page 3Linked to original sources
arXiv2022

Post-Hopf algebras, relative Rota-Baxter operators and solutions of the Yang-Baxter equation

In this paper, first we introduce the notion of a post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman-Larsson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions of the Yang-Baxter equation. Then we introduce the notion of relative Rota-Baxter operators on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota-Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota-Baxter operator also induces a post-Hopf algebra. Then we show that relative Rota-Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota-Baxter operators give solutions of the Yang-Baxter equation in certain cocommutative Hopf algebras. Finally we characterize relative Rota-Baxter operators on Hopf algebras using relative Rota-Baxter operators on the Lie algebra of primitive elements, graphs and module bialgebra structures.

math-ph↗
arXiv2022

$F$-algebroids and deformation quantization via pre-Lie algebroids

In this paper, first we introduce a new approach to the notion of $F$-algebroids, which is a generalization of $F$-manifold algebras and $F$-manifolds, and show that $F$-algebroids are the corresponding semi-classical limits of pre-Lie formal deformations of commutative associative algebroids. Then we use the deformation cohomology of pre-Lie algebroids to study pre-Lie infinitesimal deformations and extension of pre-Lie $n$-deformations to pre-Lie $(n+1)$-deformations of a commutative associative algebroid. Next we develop the theory of Dubrovin's dualities of $F$-algebroids with eventual identities and use Nijenhuis operators on $F$-algebroids to construct new $F$-algebroids. Finally we introduce the notion of pre-$F$-algebroids, which is a generalization of $F$-manifolds with compatible flat connections. Dubrovin's dualities of pre-$F$-algebroids with eventual identities, Nijenhuis operators on pre-$F$-algebroids and their applications to integral systems are discussed.

math-ph↗
arXiv2022

Deformations, cohomologies and integrations of relative difference Lie algebras

In this paper, first using the higher derived brackets, we give the controlling algebra of relative difference Lie algebras, which are also called crossed homomorphisms or differential Lie algebras of weight 1 when the action is the adjoint action. Then using Getzler's twisted $L_\infty$-algebra, we define the cohomology of relative difference Lie algebras. In particular, we define the regular cohomology of difference Lie algebras by which infinitesimal deformations of difference Lie algebras are classified. We also define the cohomology of difference Lie algebras with coefficients in arbitrary representations, and using the second cohomology group to classify abelian extensions of difference Lie algebras. Finally, we show that any relative difference Lie algebra can be integrated to a relative difference Lie group in a functorial way.

math.RA↗
arXiv2022

Actions of monoidal categories and representations of Cartan type Lie algebras

Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is established to give new weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs). This generalizes and unifies various existing constructions of representations of many Lie algebras by using this new bifunctor. We construct some crossed homomorphisms in different situations and use our actions of monoidal categories to recover some known constructions of representations of various Lie algebras, also to obtain new representations for generalized Witt algebras and their Lie subalgebras. The cohomology theory of crossed homomorphisms between Lie algebras is introduced and used to study linear deformations of crossed homomorphisms.

math.RT↗
arXiv2021

Crossed homomorphisms and Cartier-Kostant-Milnor-Moore theorem for difference Hopf algebras

The celebrated Milnor-Moore theorem and the more general Cartier-Kostant-Milnor-Moore theorem establish close relationships of a connected and a pointed cocommutative Hopf algebra with its Lie algebra of primitive elements and its group of group-like elements. Crossed homomorphisms for Lie algebras, groups and Hopf algebras have been studied extensively, first from a cohomological perspective and then more broadly, with an important case given by difference operators. This paper shows that the relationship among the different algebraic structures captured in the Milnor-Moore theorem can be strengthened to include crossed homomorphisms and differenece operators. We give a graph characterization of Hopf algebra crossed homomorphisms which are also compatible with the Milnor-Moore relation. We further investigate derived actions from crossed homomorphisms on groups, Lie algebras and Hopf algebras, and establish their relationship. Finally we obtain a Cartier-Kostant-Milnor-Moore type structure theorem for pointed cocommutative difference Hopf algebras. Examples and classifications of difference operators are also provided for several Hopf algebras.

math.QA↗
arXiv2021

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.

math-ph↗
arXiv2021

Reynolds $n$-Lie algebras and NS-$n$-Lie algebras

In this paper, first we introduce the notion of a Reynolds operator on an $n$-Lie algebra and illustrate the relationship between Reynolds operators and derivations on an $n$-Lie algebra. We give the cohomology theory of Reynolds operators on an $n$-Lie algebra and study infinitesimal deformations of Reynolds operators using the second cohomology group. Then we introduce the notion of NS-$n$-Lie algebras, which are generalizations of both $n$-Lie algebras and $n$-pre-Lie algebras. We show that an NS-$n$-Lie algebra gives rise to an $n$-Lie algebra together with a representation on itself. Reynolds operators and Nijenhuis operators on an $n$-Lie algebra naturally induce NS-$n$-Lie algebra structures. Finally, we construct Reynolds $(n+1)$-Lie algebras and Reynolds $3$-Lie algebras from Reynolds $n$-Lie algebras and Reynolds commutative associative algebras respectively.

math-ph↗
arXiv2021

Representations and cohomologies of relative Rota-Baxter Lie algebras and applications

In this paper, first we give the notion of a representation of a relative Rota-Baxter Lie algebra and introduce the cohomologies of a relative Rota-Baxter Lie algebra with coefficients in a representation. Then we classify abelian extensions of relative Rota-Baxter Lie algebras using the second cohomology group, and classify skeletal relative Rota-Baxter Lie 2-algebras using the third cohomology group as applications. At last, using the established general framework of representations and cohomologies of relative Rota-Baxter Lie algebras, we give the notion of representations of Rota-Baxter Lie algebras, which is consistent with representations of Rota-Baxter associative algebras in the literature, and introduce the cohomologies of Rota-Baxter Lie algebras with coefficients in a representation. Applications are also given to classify abelian extensions of Rota-Baxter Lie algebras and skeletal Rota-Baxter Lie 2-algebras.

math.RT↗
arXiv2021

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

In this paper, first we introduce the notion of a quadratic Lie-Yamaguti algebra and 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.

math.RA↗
arXiv2021

Twisted Rota-Baxter operators on 3-Lie algebras and NS-3-Lie algebras

In this paper, first we introduce the notion of a twisted Rota-Baxter operator on a 3-Lie algebra $\g$ with a representation on $V$. We show that a twisted Rota-Baxter operator induces a 3-Lie algebra structure on $V$, which represents on $\g$. By this fact, we define the cohomology of a twisted Rota-Baxter operator and study infinitesimal deformations of a twisted Rota-Baxter operator using the second cohomology group. Then we introduce the notion of an NS-3-Lie algebra, which produces a 3-Lie algebra with a representation on itself. We show that a twisted Rota-Baxter operator induces an NS-3-Lie algebra naturally. Thus NS-3-Lie algebras can be viewed as the underlying algebraic structures of twisted Rota-Baxter operators on 3-Lie algebras. Finally we show that a Nijenhuis operator on a 3-Lie algebra gives rise to a representation of the deformed 3-Lie algebra and a 2-cocycle. Consequently, the identity map will be a twisted Rota-Baxter operator on the deformed 3-Lie algebra. We also introduce the notion of a Reynolds operator on a 3-Lie algebra, which can serve as a special case of twisted Rota-Baxter operators on 3-Lie algebras.

math.RA↗
arXiv2021

Maurer-Cartan characterizations and cohomologies of compatible Lie algebras

In this paper, we give Maurer-Cartan characterizations as well as a cohomology theory for compatible Lie algebras. Explicitly, we first introduce the notion of a bidifferential graded Lie algebra and thus give Maurer-Cartan characterizations of compatible Lie algebras. Then we introduce a cohomology theory of compatible Lie algebras and use it to classify infinitesimal deformations and abelian extensions of compatible Lie algebras. In particular, we introduce the reduced cohomology of a compatible Lie algebra and establish the relation between the reduced cohomology of a compatible Lie algebra and the cohomology of the corresponding compatible linear Poisson structures introduced by Dubrovin and Zhang in their study of bi-Hamiltonian structures. Finally, we use the Maurer-Cartan approach to classify nonabelian extensions of compatible Lie algebras.

math-ph↗
arXiv2021

The controlling $L_\infty$-algebra, cohomology and homotopy of embedding tensors and Lie-Leibniz triples

In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and an $L_\infty$-algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie-Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$_\infty$-algebra. We realize Kotov and Strobl's construction of an $L_\infty$-algebra from an embedding tensor, to a functor from the category of homotopy embedding tensors to that of Leibniz$_\infty$-algebras, and a functor further to that of $L_\infty$-algebras.

math-ph↗
arXiv2020

Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation

In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this Hexp map. We also describe a Hom-Lie group action on a smooth manifold. Subsequently, we give the notion of an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie algebra $(\mathfrak{gl}(V),[\cdot,\cdot],\mathsf{Ad})$, and the derivation Hom-Lie algebra of a Hom-Lie algebra.

math.DG↗
arXiv2020

Twilled 3-Lie algebras, generalized matched pairs of 3-Lie algebras and O-operators

In this paper, first we introduce the notion of a twilled 3-Lie algebra, and construct an $L_\infty$-algebra, whose Maurer-Cartan elements give rise to new twilled 3-Lie algebras by twisting. In particular, we recover the Lie $3$-algebra whose Maurer-Cartan elements are O-operators (also called relative Rota-Baxter operators) on 3-Lie algebras. Then we introduce the notion of generalized matched pairs of 3-Lie algebras using generalized representations of 3-Lie algebras, which will give rise to twilled 3-Lie algebras. The usual matched pairs of 3-Lie algebras correspond to a special class of twilled 3-Lie algebras, which we call strict twilled 3-Lie algebras. Finally, we use O-operators to construct explicit twilled 3-Lie algebras, and explain why an $r$-matrix for a 3-Lie algebra can not give rise to a double construction 3-Lie bialgebra. Examples of twilled 3-Lie algebras are given to illustrate the various interesting phenomenon.

math.RA↗
arXiv2020

Integration and geometrization of Rota-Baxter Lie algebras

This paper first introduces the notion of a Rota-Baxter operator (of weight $1$) on a Lie group so that its differentiation gives a Rota-Baxter operator on the corresponding Lie algebra. Direct products of Lie groups, including the decompositions of Iwasawa and Langlands, carry natural Rota-Baxter operators. Formal inverse of the Rota-Baxter operator on a Lie group is precisely the crossed homomorphism on the Lie group, whose tangent map is the differential operator of weight $1$ on a Lie algebra. A factorization theorem of Rota-Baxter Lie groups is proved, deriving directly on the Lie group level, the well-known global factorization theorems of Semenov-Tian-Shansky in his study of integrable systems. As geometrization, the notions of Rota-Baxter Lie algebroids and Rota-Baxter Lie groupoids are introduced, with the former a differentiation of the latter. Further, a Rota-Baxter Lie algebroid naturally gives rise to a post-Lie algebroid, generalizing the well-known fact for Rota-Baxter Lie algebras and post-Lie algebras. It is shown that the geometrization of a Rota-Baxter Lie algebra or a Rota-Baxter Lie group can be realized by its action on a manifold. Examples and applications are provided for these new notions.

math.QA↗
arXiv2020

Deformations and homotopy theory of relative Rota-Baxter Lie algebras

We determine the \emph{$L_\infty$-algebra} that controls deformations of a relative Rota-Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying LieRep pair by the dg Lie algebra controlling deformations of the relative Rota-Baxter operator. Consequently, we define the {\em cohomology} of relative Rota-Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota-Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a \emph{homotopy} relative Rota-Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota-Baxter Lie algebras is intimately related to \emph{pre-Lie$_\infty$-algebras}.

math.QA↗
arXiv2020

Categorification of VB-Lie algebroids and VB-Courant algebroids

In this paper, first we introduce the notion of a $\VB$-Lie $2$-algebroid, which can be viewed as the categorification of a $\VB$-Lie algebroid. The tangent prolongation of a Lie $2$-algebroid is a $\VB$-Lie $2$-algebroid naturally. We show that after choosing a splitting, there is a one-to-one correspondence between $\VB$-Lie $2$-algebroids and flat superconnections of a Lie 2-algebroid on a 3-term complex of vector bundles. Then we introduce the notion of a $\VB$-$\LWX$ 2-algebroid, which can be viewed as the categorification of a $\VB$-Courant algebroid. We show that there is a one-to-one correspondence between split Lie 3-algebroids and split $\VB$-$\LWX$ 2-algebroids. The notion of a $\VB$-Lie $2$-bialgebroid is introduced and the double of a $\VB$-Lie $2$-bialgebroid is a $\VB$-$\LWX$ 2-algebroid. Finally, we introduce the notion of an $E$-$\LWX$ 2-algebroid and show that associated to a $\VB$-$\LWX$ 2-algebroid, there is an $E$-$\LWX$ 2-algebroid structure on the graded fat bundle naturally. By this result, we give a construction of a Lie 3-algebra from a given Lie 3-algebra, which provides interesting examples of Lie 3-algebras including the higher analogue of the string Lie 2-algebra.

math-ph↗