Searcharxiv⌕ Search

arXiv subjects

Rong Tang

Publications and source records attributed to Rong Tang.

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

Minimax Rate of Distribution Estimation on Unknown Submanifold under Adversarial Losses

Statistical inference from high-dimensional data with low-dimensional structures has recently attracted lots of attention. In machine learning, deep generative modeling approaches implicitly estimate distributions of complex objects by creating new samples from the underlying distribution, and have achieved great success in generating synthetic realistic-looking images and texts. A key step in these approaches is the extraction of latent features or representations (encoding) that can be used for accurately reconstructing the original data (decoding). In other words, low-dimensional manifold structure is implicitly assumed and utilized in the distribution modeling and estimation. To understand the benefit of low-dimensional manifold structure in generative modeling, we build a general minimax framework for distribution estimation on unknown submanifold under adversarial losses, with suitable smoothness assumptions on the target distribution and the manifold. The established minimax rate elucidates how various problem characteristics, including intrinsic dimensionality of the data and smoothness levels of the target distribution and the manifold, affect the fundamental limit of high-dimensional distribution estimation. To prove the minimax upper bound, we construct an estimator based on a mixture of locally fitted generative models, which is motivated by the partition of unity technique from differential geometry and is necessary to cover cases where the underlying data manifold does not admit a global parametrization. We also propose a data-driven adaptive estimator that is shown to simultaneously attain within a logarithmic factor of the optimal rate over a large collection of distribution classes.

math.ST↗
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

Statistical Inference for Bayesian Risk Minimization via Exponentially Tilted Empirical Likelihood

The celebrated Bernstein von-Mises theorem ensures that credible regions from Bayesian posterior are well-calibrated when the model is correctly-specified, in the frequentist sense that their coverage probabilities tend to the nominal values as data accrue. However, this conventional Bayesian framework is known to lack robustness when the model is misspecified or only partly specified, such as in quantile regression, risk minimization based supervised/unsupervised learning and robust estimation. To overcome this difficulty, we propose a new Bayesian inferential approach that substitutes the (misspecified or partly specified) likelihoods with proper exponentially tilted empirical likelihoods plus a regularization term. Our surrogate empirical likelihood is carefully constructed by using the first order optimality condition of the empirical risk minimization as the moment condition. We show that the Bayesian posterior obtained by combining this surrogate empirical likelihood and the prior is asymptotically close to a normal distribution centering at the empirical risk minimizer with covariance matrix taking an appropriate sandwiched form. Consequently, the resulting Bayesian credible regions are automatically calibrated to deliver valid uncertainty quantification. Computationally, the proposed method can be easily implemented by Markov Chain Monte Carlo sampling algorithms. Our numerical results show that the proposed method tends to be more accurate than existing state-of-the-art competitors.

stat.ME↗
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

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

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

Deformations of relative Rota-Baxter operators on Leibniz algebras

In this paper, we introduce the cohomology theory of relative Rota-Baxter operators on Leibniz algebras. We use the cohomological approach to study linear and formal deformations of relative Rota-Baxter operators. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order n deformations of a relative Rota-Baxter operator are also characterized in terms of the cohomology theory.

math.RA↗
arXiv2020

Leibniz bialgebras, relative Rota-Baxter operators and the classical Leibniz Yang-Baxter equation

In this paper, first we introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras are equivalent. Then we introduce the notion of a (relative) Rota-Baxter operator on a Leibniz algebra and construct the graded Lie algebra that characterizes relative Rota-Baxter operators as Maurer-Cartan elements. By these structures and the twisting theory of twilled Leibniz algebras, we further define the classical Leibniz Yang-Baxter equation, classical Leibniz r-matrices and triangular Leibniz bialgebras. Finally, we construct solutions of the classical Leibniz Yang-Baxter equation using relative Rota-Baxter operators and Leibniz-dendriform algebras.

math-ph↗
arXiv2019

Review of deformation theory I: Concrete formulas for deformations of algebraic structures

In this review article, first we give the concrete formulas of representations and cohomologies of associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues. Then we recall the graded Lie algebras and graded associative algebras that characterize these algebraic structures as Maurer-Cartan elements. The corresponding Maurer-Cartan element equips the graded Lie or associative algebra with a differential. Then the deformations of the given algebraic structures are characterized as the Maurer-Cartan elements of the resulting differential graded Lie or associative algebras. We also recall the relation between the cohomologies and the differential graded Lie and associative algebras that control the deformations.

math.RA↗
arXiv2019

Review of deformation theory II: a homotopical approach

We give a general treatment of deformation theory from the point of view of homotopical algebra following Hinich, Manetti and Pridham. In particular, we show that any deformation functor in characteristic zero is controlled by a certain differential graded Lie algebra defined up to homotopy, and also formulate a noncommutative analogue of this result valid in any characteristic.

math.AT↗
arXiv2019

Homotopy Rota-Baxter operators, homotopy $\mathcal{O}$-operators and homotopy post-Lie algebras

Rota-Baxter operators, $\mathcal{O}$-operators on Lie algebras and their interconnected pre-Lie and post-Lie algebras are important algebraic structures with applications in mathematical physics. This paper introduces the notions of a homotopy Rota-Baxter operator and a homotopy $\mathcal{O}$-operator on a symmetric graded Lie algebra. Their characterization by Maurer-Cartan elements of suitable differential graded Lie algebras is provided. Through the action of a homotopy $\mathcal{O}$-operator on a symmetric graded Lie algebra, we arrive at the notion of an operator homotopy post-Lie algebra, together with its characterization in terms of Maurer-Cartan elements. A cohomology theory of post-Lie algebras is established, with an application to 2-term skeletal operator homotopy post-Lie algebras.

math.QA↗
arXiv2019

Cohomologies of a Lie algebra with a derivation and applications

The main object of study of this paper is the notion of a LieDer pair, i.e. a Lie algebra with a derivation. We introduce the concept of a representation of a LieDer pair and study the corresponding cohomologies. We show that a LieDer pair is rigid if the second cohomology group is trivial, and a deformation of order n is extensible if its obstruction class, which is defined to be an element is the third cohomology group, is trivial. We classify central extensions of LieDer pairs using the second cohomology group with the coefficient in the trivial representation. For a pair of derivations, we define its obstruction class and show that it is extensible if and only if the obstruction class is trivial. Finally, we classify Lie2Der pairs using the third cohomology group of a LieDer pair.

math.RT↗
arXiv2019

Representations and cohomologies of Hom-pre-Lie algebras

In this paper, first we study dual representations and tensor representations of Hom-pre-Lie algebras. Then we develop the cohomology theory of Hom-pre-Lie algebras in term of the cohomology theory of Hom-Lie algebras. As applications, we study linear deformations of Hom-pre-Lie algebras, which are characterized by the second cohomology groups of Hom-pre-Lie algebras with the coefficients in the regular representation. The notion of a Nijenhuis operator on a Hom-pre-Lie algebra is introduced which can generate trivial linear deformations of a Hom-pre-Lie algebra. Finally, we introduce the notion of a Hessian structure on a Hom-pre- Lie algebra, which is a symmetric nondegenerate 2-cocycle with the coefficient in the trivial representation. We also introduce the notion of an $\huaO$-operator on a Hom-pre-Lie algebra, by which we give an equivalent characterization of a Hessian structure.

math.RA↗
arXiv2018

Generalized metric $n$-Leibniz algebras and generalized orthogonal representation of metric Lie algebras

We introduce the notion of a generalized metric n-Leibniz algebra and show that there is a one-to-one correspondence between generalized metric n-Leibniz algebras and faithful generalized orthogonal representations of metric Lie algebras (called Lie triple data). We further show that there is also a one-to-one correspondence between generalized orthogonal derivations (resp. generalized orthogonal automorphisms) on generalized metric n-Leibniz algebras and Lie triple datas.

math.RA↗
arXiv2018

Deformations and their controlling cohomologies of $\mathcal{O}$-operators

$\mathcal{O}$-operators are important in broad areas in mathematics and physics, such as integrable systems, the classical Yang-Baxter equation, pre-Lie algebras and splitting of operads. In this paper, a deformation theory of $\mathcal{O}$-operators is established in consistence with the general principles of deformation theories. On the one hand, $\mathcal{O}$-operators are shown to be characterized as the Maurer-Cartan elements in a suitable graded Lie algebra. A given $\mathcal{O}$-operator gives rise to a differential graded Lie algebra whose Maurer-Cartan elements characterize deformations of the given $\mathcal{O}$-operator. On the other hand, a Lie algebra with a representation is identified from an $\mathcal{O}$-operator $T$ such that the corresponding Chevalley-Eilenberg cohomology controls deformations of $T$, thus can be regarded as an analogue of the André-Quillen cohomology for the $\mathcal{O}$-operator. Thereafter, infinitesimal and formal deformations of $\mathcal{O}$-operators are studied. In particular, the notion of Nijenhuis elements is introduced to characterize trivial infinitesimal deformations. Formal deformations and extendibility of order $n$ deformations of an $\mathcal{O}$-operator are also characterized in terms of the new cohomology theory. Applications are given to deformations of Rota-Baxter operators of weight 0 and skew-symmetric $r$-matrices for the classical Yang-Baxter equation. For skew-symmetric $r$-matrices, there is an independent Maurer-Cartan characterization of the deformations as well as an analogue of the André-Quillen cohomology controlling the deformations, which turn out to be equivalent to the ones obtained as $\mathcal{O}$-operators associated to the coadjoint representations. Finally, infinitesimal deformations of skew-symmetric $r$-matrices and their corresponding triangular Lie bialgebras are studied.

math.QA↗