Hua-Lin Huang, Gongxiang Liu, Yu Ye
Let $k$ be an algebraically closed field of characteristic $p>0$. For a loop $\circlearrowleft$, denote its path coalgebra by $k\circlearrowleft$. In this paper, all the finite-dimensional commutative Hopf algebras over the sub coalgebras of $k\circlearrowleft$ are given. As a direct consequence, all the commutative infinitesimal groups $\mathcal{G}$ with dim$_{k}$Lie$(\mathcal{G})=1$ are classified.
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Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^\circ$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^\circ$-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner-Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of $A^\circ$ in $R^A$.
In this paper we prove that if $R$ is a commutative, reduced, local ring, then $R$ is Hopfian if and only if the ring $R[x]$ is Hopfian. This answers a question of Varadarajan, in the case when $R$ is a reduced local ring. We provide examples of non-Noetherian Hopfian commutative domains by proving that the finite dimensional domains are Hopfian. Also, we derive some general results related to Hopfian rings.
In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra pl(1,1), provides a noetherian prime counterexample.
A half-commutative orthogonal Hopf algebra is a Hopf *-algebra generated by the self-adjoint coefficients of an orthogonal matrix corepresentation $v=(v_{ij})$ that half commute in the sense that $abc=cba$ for any $a,b,c \in \{v_{ij}\}$. The first non-trivial such Hopf algebras were discovered by Banica and Speicher. We propose a general procedure, based on a crossed product construction, that associates to a self-transpose compact subgroup $G \subset U_n$ a half-commutative orthogonal Hopf algebra $\mathcal A_*(G)$. It is shown that any half-commutative orthogonal Hopf algebra arises in this …
The finite dual $H^{\circ}$ of an affine commutative-by-finite Hopf algebra $H$ is studied. Such a Hopf algebra $H$ is an extension of an affine commutative Hopf algebra $A$ by a finite dimensional Hopf algebra $F$. The main theorem gives natural conditions under which $H^{\circ}$ decomposes as a crossed or smash product of $F^{\ast}$ by the finite dual of $A$. This decomposition is then further analysed using the Cartier- Gabriel-Kostant theorem to obtain component Hopf subalgebras of $H^{\circ}$ mapping onto the classical components of $A^{\circ}$. The detailed consequences for a number of f…
The objects of study in this paper are Hopf algebras $H$ which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra by a finite dimensional Hopf algebra. Basic structural and homological properties are recalled and classes of examples are listed. Bounds are obtained on the dimensions of simple $H$-modules, and the structure of $H$ is shown to be severely constrained when the finite dimensional extension is semisimple and cosemisimple.
In this paper, we begin a systematic study of modified Rota-Baxter algebras, as an associative analogue of the modified classical Yang-Baxter equation. We construct free commutative modified Rota-Baxter algebras by a variation of the shuffle product and describe the structure both recursively and explicitly. We then provide these algebras with a Hopf algebra structure by applying a Hochschild cocycle.
In this work, the cohomology theory for partial actions of co-commutative Hopf algebras over commutative algebras is formulated. This theory generalizes the cohomology theory for Hopf algebras introduced by Sweedler and the cohomology theory for partial group actions, introduced by Dokuchaev and Khrypchenko. Some nontrivial examples, not coming from groups are constructed. Given a partial action of a co-commutative Hopf algebra $H$ over a commutative algebra $A$, we prove that there exists a new Hopf algebra $\widetilde{A}$, over a commutative ring $E(A)$, upon which $H$ still acts partially a…
The so called induction functors appear in several areas of Algebra in different forms. Interesting examples are the induction functors in the Theory of Affine Algebraic groups. In this note we investigate the so called Hopf pairings (bialgebra pairings) and use them to study induction functors for affine group schemes over arbitrary commutative ground rings. We present also a special type of Hopf pairings (bialgebra pairings), satisfying the so called $α$-condition. For those pairings the induction functor is studied and a nice description of it is obtained. Along the paper several interestin…
We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of looking for bases which generate the space we define orders on the set partitions and set compositions which allow us to define bases which have simple and natural rules for the product of basis elements.
We consider the combinatorial Dyson-Schwinger equation X=B^+(P(X)) in the non-commutative Connes-KreimerHopf algebra of planar rooted trees H, where B^+ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra A_P of\H. We describe all the formal series P such that A_P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of H, organized into three isomorphism classes: a first one, restricted to a olynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, i…
Motivated by the Hopf algebra structures established on free commutative Rota-Baxter algebras, we explore Hopf algebra related structures on free commutative Nijenhuis algebras. Applying a cocycle condition, we first prove that a free commutative Nijenhuis algebra on a left counital bialgebra (in the sense that the right-sided counicity needs not hold) can be enriched to a left counital bialgebra. We then establish a general result that a connected graded left counital bialgebra is a left counital right antipode Hopf algebra in the sense that the antipode is also only right-sided. We finally a…
Let H be a semisimple (so, finite dimensional) Hopf algebra over an algebraically closed field k of characteristic zero and let A be a commutative domain over k. We show that if A arises as an H-module algebra via an inner faithful H-action, then H must be a group algebra. This answers a question of E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen. The main results of this article extend to working over k of positive characteristic. On the other hand, we obtain results on Hopf actions on Weyl algebras as a consequence of the main theorem.
To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra morphism taking its values in Q[X], we deduce from the interacting bialgebras an algebraic proof of the duality principle, a generalization and a new proof of a result on B-series due to Whright and Zhao, using a monoid of characters on quasi-posets, and a generalization of Faulhaber's formula. We also give non-commutative versions of these results: polynomials…
Based on a pairing of two regular multiplier Hopf algebras $A$ and $B$, Heisenberg double $\mathscr{H}$ is the smash product $A \# B$ with respect to the left regular action of $B$ on $A$. Let $\mathscr{D}=A\bowtie B$ be the Drinfel'd double, then Heisenberg double $\mathscr{H}$ is a Yetter-Drinfel'd $\mathscr{D}$-module algebra, and it is also braided commutative by the braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to some infinite dimensional cases.
In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple non-commutative Hopf algebras of dimension 16. Moreover, we prove that non-commutative semisimple Hopf algebras of dimension p^n, p is prime, cannot have a cyclic group of grouplikes.
Post-Hopf algebra in non-commutative probability theory
该研究探讨了李代数上相容的$\mathcal{O}$-算子与后李乘积的群乘积分解问题，并将其应用于非交换概率论，通过引入新变换简化了算子值非交换分布的条件自由和条件单调乘法卷积计算。
We generalize the Wiener-Hopf factorization of Laurent series to more general commutative coefficient rings, and we give explicit formulas for the decomposition. We emphasize the algebraic nature of this factorization.
Let $L/K$ be a Galois extension of fields with Galois group $G$, an elementary abelian $p$-group of rank $n$ for $p$ an odd prime. It is known that nilpotent $\mathbb{F}_p$-algebra structures $A$ on $G$ yield regular subgroups of the holomorph of $G$, hence Hopf Galois structures on $L/K$. In this paper we illustrate the richness of Hopf Galois structures on $L/K$ by examining the case where $A$ is abelian of dimension $n$ where the dimension of $A^2 = 1$. We determine the number of Hopf Galois structures that arise in these cases, describe those structures explicitly, and estimate the extent …
The theory of operated algebras has played a pivotal role in mathematics and physics. In this paper, we introduce a $λ$-TD algebra that appropriately includes both the Rota-Baxter algebra and the TD-algebra. The explicit construction of free commutative $λ$-TD algebra on a commutative algebra is obtained by generalized shuffle products, called $λ$-TD shuffle products. We then show that the free commutative $λ$-TD algebra possesses a left counital bialgera structure by means of a suitable 1-cocycle condition. Furthermore, the classical result that every connected filtered bialgebra is a Hopf al…
For any graded bialgebras $A$ and $B$, we define a commutative graded algebra $A_B$ representing the functor of $B$-representations of $A$. When $A$ is a cocommutative graded Hopf algebra and $B$ is a commutative ungraded Hopf algebra, we introduce a method deriving a Gerstenhaber bracket in $A_B$ from a Fox pairing in $A$ and a balanced biderivation in $B$. Our construction is inspired by Van den Bergh's non-commutative Poisson geometry, and may be viewed as an algebraic generalization of the Atiyah--Bott--Goldman Poisson structures on moduli spaces of representations of surface groups.
Smoothness of commutative Hopf algebras
本文研究了半单阿贝尔对称幺单范畴中交换Hopf代数的等变光滑性，证明了代数光滑性、余模代数光滑性与积态射单射性的等价条件，并分析了特征零与正特征下各类范畴中Hopf代数的光滑性质。
Many related products and coproducts (e.g. Hadamard, Cauchy, Kronecker, induction, internal, external, Solomon, composition, Malvenuto-Reutenauer, convolution, etc.) have been defined in the following objects : species, representations of the symmetric groups, symmetric functions, endomorphisms of graded connected Hopf algebras, permutations, non-commutative symmetric functions, quasi-symmetric functions, etc. With the purpose of simplifying and unifying this diversity we introduce yet, another -non graded- product the Heisenberg product, that for the highest and lowest degrees produces the cl…
In this paper, we introduce and investigate \emph{bisemialgebras}and\emph{\ Hopf semialgebras} over commutative semirings. We generalize to the semialgebraic context several results on bialgebras and Hopf algebras over rings including the main reconstruction theorems and the \emph{Fundamental Theorem of Hopf Algebras}. We also provide a notion of \emph{quantum monoids} as Hopf semialgebras which are neither commutative nor cocommutative; this extends the Hopf algebraic notion of a quantum group. The generalization to the semialgebraic context is neither trivial nor straightforward due to the n…
We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables. We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various…
Let $A$ be a finite commutative nilpotent $\mathbb{F}_p$-algebra structure on $G$, an elementary abelian group of order $p^n$. If $K/k$ is a Galois extension of fields with Galois group $G$ and $A^p = 0$, then corresponding to $A$ is an $H$-Hopf Galois structure on $K/k$ of type $G$. For that Hopf Galois structure we may study the image of the Galois correspondence from $k$-subHopf algebras of $H$ to subfields of $K$ containing $k$ by utilizing the fact that the intermediate subfields correspond to the $\mathbb{F}_p$-subspaces of $A$, while the subHopf algebras of $H$ correspond to the ideals …
Rota-Baxter systems of Hopf algebras and Hopf trusses
本文提出Hopf代数的Rota-Baxter系统概念，探究其与Rota-Baxter Hopf代数的关系、群代数上的对应结构、派生Hopf代数及特征群的局部分解。
We develop Lie's correspondence and an explicit Baker-Campbell-Hausdorff formula for commutative automorphic formal loops.
On $4n$-dimensional neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras
本文研究了 $4n$ 维非指针非半单霍普夫代数及其相关弱霍普夫代数的结构，揭示了其准三角性、 $R$-矩阵描述及绿色环的非可交换性。
The various finiteness conditions in commutative Moufang loops are characterized using the notions of centralizer of subloops and centralizer of subgroups of its multiplication group.
It is proved that the maximum condition for subloops in a commutative Moufang loop $Q$ is equivalent with the conditions of finite generating of different subloops of the loop $Q$ and different subgroups of the multiplication group of the loop $Q$. An analogue equivalence is set for the commutative Moufang $ZA$-loops.
Hopf algebras and associative representations of two-dimensional evolution algebras
本文通过自同构代数群建立二维演化代数与Hopf代数的联系，计算了相关联的p-代数，并给出了忠实交换结合表示存在性及同构条件的刻画。
This paper is a written form of a talk. It gives a review of various notions of Galois (and in particular cleft) extensions. Extensions by coalgebras,bialgebras and Hopf algebras (over a commutative base ring) and by corings,bialgebroids and Hopf algebroids (over a non-commutative base algebra) are systematically recalled and compared. In the first version of this paper, the journal version of [15, Theorem 2.6] was heavily used, in two respects. First, it was applied to establish an isomorphism between the comodule categories of two constituent bialgebroids in a Hopf algebroid. Second, it was …
Wick polynomials and Wick products are studied in the context of non-commutative probability theory. It is shown that free, boolean and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf algebraic approach to cumulants and Wick products in classical probability theory.
We show that the functor of $p$-typical co-Witt vectors on commutative algebras over a perfect field $k$ of characteristic $p$ is defined on, and in fact only depends on, a weaker structure than that of a $k$-algebra. We call this structure a $p$-polar $k$-algebra. By extension, the functors of points for any $p$-adic affine commutative group scheme and for any formal group are defined on, and only depend on, $p$-polar structures. In terms of abelian Hopf algebras, we show that a cofree cocommutative Hopf algebra can be defined on any $p$-polar $k$-algebra $P$, and it agrees with the cofree co…
In his generalization of reductive homogeneous spaces, Lev Sabinin showed that Lie's fundamental theorems hold for local analytic hyporeductive and pseudoreductive loops. We derive Sabinin's results in an algebraic context in terms of non-associative Hopf algebras that satisfy the analog of the hyporeductive and pseudoreductive identities for loops.
We consider graded representations of the algebra NC of noncommutative symmetric functions on the Z-linear span of a graded poset P. The matrix coefficients of such a representation give a Hopf morphism from a Hopf algebra HP generated by the intervals of P to the Hopf algebra of quasi-symmetric functions. This provides a unified construction of quasi-symmetric generating functions from different branches of algebraic combinatorics, and this construction is useful for transferring techniques and ideas between these branches. In particular we show that the (Hopf) algebra of Billera and Liu rela…
This is a study on pattern Hopf algebras in combinatorial structures. We introduce the notion of combinatorial presheaf, by adapting the algebraic framework of species to the study of substructures in combinatorics. Afterwards, we consider functions that count the number of patterns of objects and endow the linear span of these functions with a product and a coproduct. In this way, any well behaved family of combinatorial objects that admits a notion of substructure generates a Hopf algebra, and this association is functorial. For example, the Hopf algebra on permutations studied by Vargas in …
We study the Yetter--Drinfeld D(B)-module algebra structure on the Heisenberg double H(B^*) endowed with a "heterotic" action of the Drinfeld double D(B). This action can be interpreted in the spirit of Lu's description of H(B^*) as a twist of D(B). In terms of the braiding of Yetter--Drinfeld modules, H(B^*) is braided commutative. By the Brzezinski--Militaru theorem, H(B^*)#D(B) is then a Hopf algebroid over H(B^*). For B a particular Taft Hopf algebra at a 2p-th root of unity, the construction is adapted to yield Yetter--Drinfeld module algebras over the 2p^3-dimensional quantum group U_q(s…
Let $k$ be a commutative ring, $H$ a faithfully flat Hopf algebra with bijective antipode, $A$ a $k$-flat right $H$-comodule algebra. We investigate when a relative Hopf module is projective over the subring of coinvariants $B=A^{{\rm co}H}$, and we study the semisimplicity of the category of relative Hopf modules.
On the structure of abelian Hopf algebras
本文研究特征 $p$ 完美域上分次连通交换余交换霍普夫代数的结构，给出 $p$-挠对象的不可分解直和分解，并分类自由或余自由对象及模 $p$ 不可分解对象。
On the Semi-Abelianness of Affine Group Schemes
本文证明了在域$k$上交换Hopf代数的范畴是co-semi-abelian的，并推导出仿射群$k$-方案的范畴是semi-abelian的。
The aim of this paper is to establish a duality between the category of discrete groupoids and the category of geometrically transitive commutative Hopf algebroids in the sense of P. Deligne and A. Bruguières. In one direction we have the usual contravariant functor which assigns to each Hopf algebroid its characters groupoid (the fiber groupoid at the ground field). In the other direction we construct the contravariant functor which associated to each discrete groupoid its Hopf algebroid of representative functions. This duality extends the well known duality between discrete groups and commu…
Actions of semisimple Hopf algebras H over an algebraically closed field of characteristic zero on commutative domains were classified recently by the authors. The answer turns out to be very simple- if the action is inner faithful, then H has to be a group algebra. The present article contributes to the non-semisimple case, which is much more complicated. Namely, we study actions of finite dimensional (not necessarily semisimple) Hopf algebras on commutative domains, particularly when H is pointed of finite Cartan type. The work begins by reducing to the case where H acts inner faithfully on …
A Noetherian Hopf algebra is affine iff its Hopf coradical is affine
本文通过引入非交换约化序、因子分解理论和广义提升方法，建立了诺特Hopf代数仿射性的充要条件，即左或右诺特Hopf代数仿射当且仅当其Hopf余根仿射，从而推进了Wu-Zhang问题的解决。
This paper introduces Hopf braces, a new algebraic structure related to the Yang-Baxter equation which include Rump's braces and their non-commutative generalizations as particular cases. Several results of classical braces are still valid in our context. Furthermore, Hopf braces provide the right setting for considering left symmetric algebras as Lie-theoretical analogs of braces.
We compute the orders of free commutative Moufand loops of exponent 3 with $n\leq 7$ free generators and find embeddings of such loops into a loop of invertible elements of the free commutative alternative algebra with identity $x^3=0$.
Towards differentiation and integration between Hopf algebroids and Lie algebroids
本文在交换 Hopf algebroids 与 Lie-Rinehart algebras 背景下，构造了形式微分与积分的反变函子，揭示了二者的伴随关系，并利用微分函子分析了 Hopf algebroids 态射的几何可分性。
The question of whether or not a Hopf algebra $H$ is faithfully flat over a Hopf subalgebra $A$ has received positive answers in several particular cases: when $H$ (or more generally, just $A$) is commutative, or cocommutative, or pointed, or when $K$ contains the coradical of $H$. We prove the statement in the title, adding the class of cosemisimple Hopf algebras to those known to be faithfully flat over all Hopf subalgebras. We also show that the third term of the resulting "exact sequence" $A\to H\to C$ is always a cosemisimple coalgebra, and that the expectation $H\to A$ is positive when $…
Let H be a connected Hopf k-algebra of finite Gel'fand-Kirillov dimension over an algebraically closed field k of characteristic 0. The objects of study in this paper are the left or right coideal subalgebras T of H. They are shown to be deformations of commutative polynomial k-algebras. A number of well-known homological and other properties follow immediately from this fact. Further properties are described, examples are considered, invariants are constructed and a number of open questions are listed.
Let A and B be two connected graded commutative k-algebras of finite type, where k is a perfect field of positive characteristic p. We prove that the quasi--shuffle algebras generated by A and B are isomorphic as Hopf algebras if and only if A and B are isomorphic as graded k-vector spaces equipped with a Frobenius (pth-power) map. For the hardest part of this analysis, we work with the dual construction, and are led to study connected graded cocommutative Hopf algebras H with two additional properties: H is free as an associative algebra, and the projection onto the indecomposables is split a…
A study of nil Hecke algebras via Hopf algebroids
本文通过 Kostant 和 Kumar 定义的余乘法，利用 Demazure 算子与单反射间的混合二面体辫关系，将与 Coxeter 系统关联的仿射 nil Hecke 代数转化为无对极的 Hopf algebroid，并附带了相关领域的实例。
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