Martin Lorenz
This note discusses a framework for the investigation of the prime spectrum of an associative algebra A that is equipped with an action of a Hopf algebra H. In particular, we study a notion of H-rationality for ideals of A and comment on a possible Dixmier-Moeglin equivalence for H-prime ideals of A.
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Hopf algebraic homogeneous spaces interpreted rationally: the Abe-Kanno theorem
将Abe-Kanno定理推广到Hopf代数，建立子Hopf代数与有理函数域不变子域之间的对应，并应用于剩余有限维Hopf代数的性质研究。
It is shown that the subalgebra of invariants of a free associative algebra of finite rank under a linear action of a semisimple Hopf algebra has a rational Hilbert series with respect to the usual degree function, whenever the ground field has zero characteristic.
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 $R^A$.
A remark on the Galois-type correspondence between ideal coideals and comodule subrings of a Hopf algebroid
本文建立了Hopf代数胚中特定左理想余理想与右余理想子环之间的双射对应，推广了作者与合作者的近期结果，并在Hopf代数情形下扩展了已知对应关系。
A Poisson algebra $\Bbb C[G]$ considered as a Poisson version of the twisted quantized coordinate ring $\Bbb C_{q,p}[G]$, constructed by Hodges, Levasseur and Toro in \cite{HoLeT}, is obtained and its Poisson structure is investigated. This establishes that all Poisson prime and primitive ideals of $\Bbb C[G]$ are characterized. Further it is shown that $\Bbb C[G]$ satisfies the Poisson Dixmier-Moeglin equivalence and that Zariski topology on the space of Poisson primitive ideals of $\Bbb C[G]$ agrees with the quotient topology induced by the natural surjection from the maximal ideal space of …
Let $H$ be a cocommutative Hopf algebra acting on an algebra $A$. Assuming the base field to be algebraically closed and the $H$-action on $A$ to be integral, that is, it is given by a coaction of some Hopf subalgebra of the finite dual $H^\circ$ that is an integral domain, we stratify the prime spectrum $\mbox{Spec}\, A$ in terms of the prime spectra of certain commutative algebras. For arbitrary $H$-actions in characteristic $0$, we show that the largest $H$-stable ideal of $A$ that is contained in a given semiprime ideal of $A$ is semiprime as well.
The quantized enveloping algebra $U_q$ is constructed as a quotient of the generalized quantum double $ U^{\leq 0}_q \cmdbicross_τ U^{\geq 0}_q $ associated to a natural skew pairing $ τ: U^{\leq 0}_q \otimes U^{\geq 0}_q \to k $. This double is generalized by $\cmdcalD = (\cmdfrakB (V) \cmddotrtimes F) \cmdbicross_τ (\cmdfrakB (W) \cmddotrtimes G)$, where $F$, $G$ are abelian groups, $ V \in {}^F_F \cmdcalYD $, $ W \in {}^G_G \cmdcalYD $ are Yetter-Drinfeld modules and $ \cmdfrakB (V) $, $ \cmdfrakB (W) $ are their Nichols algebras. We prove some results on Hopf ideals of $ \cmdcalD $, includ…
Pointed Hopf algebra (co)actions on rational functions
本文提出通过代数同态构造作用于给定代数的Hopf代数的方法，可控制作用限制、点Hopf代数性质及$*$-结构相容性，并将其应用于有理函数域，得到可延拓到有理函数的尖点量子齐性空间结构。
Let $H$ be a finite-dimensional Hopf algebra. We study the behaviou r of primitive and maximal ideals in certain types of ring extensions determined by $H$. The main focus is on the class of faithfully flat Galois extensions, which includes includes smash and crossed products. It is shown how analogous results can be obtained for the larger class of extensions possessing a total integral, which includes extensions $A^H\subseteq A $ when $H$ is semisimple. We use Passman's "primitivity machine" to reduce the whole theory of Kr ull relations for prime ideals to the case of primitive ideals. The …
Let $H$ be a connected Hopf algebra acting on an algebra $A$. Working over a base field having characteristic $0$, we show that for a given prime (semi-prime, completely prime) ideal $I$ of $A$, the largest $H$-stable ideal of A contained in $I$ is also prime (semi-prime, completely prime). We also prove a similar result for certain subrings of convolution algebras.
For $p$ a prime and $a\in\mathbb{Q}$, where $a$ is not a $p^n$-th power of any rational number, the extension $\mathbb{Q}(w_n)/\mathbb{Q}$ where $w_n=\root p^n \of a$ is separable but non-normal. The Hopf-Galois theory for separable extensions was determined by Greither and Pareigis, and the specific classification for radical extensions such as these by the author. In this work we extend this theory to a certain class of profinite extensions, namely those formed from the union of these $\mathbb{Q}(w_n)$. We construct a 'profinite' Hopf algebra which acts, and show that it satisfies a generali…
Hilbert's Nullstellensatz characterizes polynomials that vanish on the vanishing set of an ideal in C[x]. In the free algebra C<X> the vanishing set of a two-sided ideal I is defined in a dimension-free way using images in finite-dimensional representations of C<x>/I. In this article Nullstellensätze for a simple but important class of ideals in the free algebra - called tentatively rationally resolvable here - are presented. An ideal is rationally resolvable if its defining relations can be eliminated by expressing some of the X variables using noncommutative rational functions in the remaini…
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 …
The lowest discriminant ideal of a Cayley-Hamilton Hopf algebra
本文研究了Cayley-Hamilton Hopf代数的最低判别理想，用极大稳定模描述其零点集，并揭示其受winding自同构群作用支配，应用于群代数及量子代数等实例。
In this paper, we study Dorroh extensions of bialgebras and Hopf algebras. Let $(H,I)$ be both a Dorroh pair of algebras and a Dorroh pair of coalgebras. We give necessary and sufficient conditions for $H\ltimes_dI$ to be a bialgebra and a Hopf algebra, respectively. We also describe all ideals of Dorroh extensions of algebras and subcoalgebras of Dorroh extensions of coalgebras and compute these ideals and subcoalgebras for some concrete examples.
On a family of liftings of the Jordan plane
研究无限循环群上Jordan平面的一族Hopf代数提升，确定其中心、素谱、本原谱和自同构群，证明每个素理想是完全素理想且非零理想与中心非平凡相交，构造对应于所有本原理想的显式单模并分类有限维单模，最后证明这些Hopf代数满足Dixmier-Moeglin等价性。
Quotients of graded quasi-Hopf algebras
研究强\(G -\)分次拟霍普夫代数的拟霍普夫理想分类，通过证明其与\((N,K)\)对应来分类，特定类拟霍普夫理想对应关系有改进，应用于余中心阿贝尔扩张等。
Correspondence theorems for Hopf algebroids with applications to affine groupoids
本文建立了bialgebroid中单边coideal子环与单边理想双边coideal的对应关系，证明其在Hopf algebroids中满足附加条件时为双射，并应用于研究交换Hopf algebroids中的正规Hopf理想与正规affine subgroupoids的联系。
Chevalley property of module-finite Hopf algebras and discriminant ideals
本文研究了Cayley-Hamilton霍夫代数的Chevalley性质，通过判别理想证明了模的完全可约性与最低判别理想的关系，并解决了Huang-Mi-Qi-Wu提出的问题。
Chevalley property and discriminant ideals of Cayley-Hamilton Hopf Algebras
研究Cayley-Hamilton霍普夫代数的Chevalley性质与判别理想，证明判别理想零集的性质及霍普夫代数的Grothendieck环的Frobenius-Perron维度。
Motivated by the study of von Neumann regular skew groups as carried out by Alfaro, Ara and del Rio in 1995 we investigate regular and biregular Hopf module algebras. If $A$ is an algebra with an action by an affine Hopf algebra $H$, then any $H$-stable left ideal of $A$ is a direct summand if and only if $A^H$ is regular and the invariance functor $(-)^H$ induces an equivalence of $A^H$-Mod to the Wisbauer category of $A$ as $A# H$-module. Analogously we show a similar statement for the biregularity of $A$ relative to $H$ where $A^H$ is replaced by $R=Z(A)\cap A^H$ using the module theory of …
In this paper, we continue our study of the tensor product structure of category $\mathcal W$ of weight modules over the Hopf-Ore extensions $kG(χ^{-1}, a, 0)$ of group algebras $kG$, where $k$ is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of $χ$ and $χ(a)$ are different. Then we describe the Green ring $r(\mathcal W)$ of the tensor category $\mathcal W$. It is shown that $r(\mathcal W)$ is isomorphic to the polynomial algebra over the group ring $\math…
We study the question of the surjectivity of the Galois correspondence from subHopf algebras to subfields given by the Fundamental Theorem of Galois Theory for abelian Hopf Galois structures on a Galois extension of fields with Galois group G', a finite abelian p-group. Applying the connection between regular subgroups of the holomorph of a finite abelian p-group (G, +) and associative, commutative nilpotent algebra structures A on (G, +) of Caranti, et. al., we show that if A gives rise to a H-Hopf Galois structure on L/K, then the K-subHopf algebras of H correspond to the ideals of A. Among …
We develop the theory of rational ideals for arbitrary associative algebras R without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur-Martindale ring of quotients replaces the classical ring of quotients which underlies the previous definition of rational ideals but is not available in a general setting. Our main result concerns rational actions of an affine algebraic group G on R. Working over an algebraically closed base field, we prove an existence and uniqueness result for generic rational ideals: for every G-rational ideal I of R, the closed…
The faithful quasi-dual $H^d$ and strict quasi-dual $H^{d'}$ of an infinite braided Hopf algebra $H$ are introduced and it is proved that every strict quasi-dual $H^{d'}$ is an $H$-Hopf module. The connection between the integrals and the maximal rational $H^{d}$-submodule $H^{d rat}$ of $H^{d}$ is found. That is, $H^{d rat}\cong \int ^l_{H^d} \otimes H$ is proved. The existence and uniqueness of integrals for braided Hopf algebras in the Yetter-Drinfeld category $(^B_B{\cal YD},C)$ are given.
"Co-Frobenius" coalgebras were introduced as dualizations of Frobenius algebras. Recently, it was shown in \cite{I} that they admit left-right symmetric characterizations analogue to those of Frobenius algebras: a coalgebra $C$ is co-Frobenius if and only if it is isomorphic to its rational dual. We consider the more general quasi-co-Frobenius (QcF) coalgebras; in the first main result we show that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or equivalently right) rational dual $Rat(C^*)$, in the sense that certain coproduct or prod…
The Dixmier-Moeglin equivalence for cocommutative Hopf algebras of finite Gelfand-Kirillov dimension
本文证明具有多项式有界增长的诺特余交换Hopf代数满足Dixmier-Moeglin等价性，并指出Lorenz的反例说明该假设不可去除，进而猜想该等价性对所有多项式有界增长的有限生成复诺特Hopf代数成立。
We define polynomial H-identities for comodule algebras over a Hopf algebra H and establish general properties for the corresponding T-ideals. In the case H is a Taft algebra or the Hopf algebra E(n), we exhibit a finite set of polynomial H-identities which distinguish the Galois objects over H up to isomorphism.
Let $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category $\mathcal C$ as modules over the smash extension $\mathcal C\# H$. We construct Grothendieck spectral sequences for the cohomologies as well as the $H$-locally finite cohomologies of these objects. We also introduce relative $(\mathcal D,H)$-Hopf modules over a Hopf comodule category $\mathcal D$. These generalize relative $(A,H)$-Hopf modules over an $H$-comodule algebra $A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational $Hom$ objects and higher derived func…
The lowest discriminant ideal of central extensions of abelian groups
本文研究任意有限群中心扩张的群代数，证明其最低判别理想的零点集为整个极大谱，并针对阿贝尔群中心扩张研究纤维代数与自同构轨道，给出同构不可提升的例子。
Let $H$ be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable $H$-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions of the tensor products of indecomposable $H$-modules by virtue of almost split sequences. The Green ring $r(H)$ of $H$ will be presented in terms of generators and relations. It turns out that the Green ring $r(H)$ is commutative and is generated by one variable over the Grothendieck ring $G_0(H)$ of $H$ modulo one relation. Moreover, $r(H)$ is Frobenius and …
Let H be a differential graded Hopf algebra over a field k. This paper gives an explicit construction of a triple cochain complex that defines the Hochschild-Cartier cohomology of H. A certain truncation of this complex is the appropriate setting for deforming H as an H(q)-structure. The direct limit of all such truncations is the appropriate setting for deforming H as a strongly homotopy associative structure. Sign complications are systematically controlled. The connection between rational perturbation theory and the deformation theory of certain free commutative differential graded algebras…
Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to prove the existence of enveloping actions, i.e., every partial Hopf action on a algebra A is induced by a Hopf action on a algebra B that contains A as a right ideal. This globalization theorem allows to extend several results from the theory of partial group actions to the Hopf algebraic setting. In this article, we prove a dual version of the globalization theorem: that every partial coaction of a Hopf algebra a…
Let $A \subset H$ be a finite centralizing extension of noetherian Hopf algebras, and let $X$ denote the group of characters of $H$ that restrict to the augmentation map on $A$. Our main results provide necessary and sufficient conditions for the fibers of the canonical surjection from $spec H$ onto $spec A$ to coincide with the $X$-orbits in $spec H$. In particular, all of the fibers are $X$-orbits if and only if the fiber over the augmentation ideal of $A$ is an $X$-orbit. An application to the representation theory of quantum function algebras, at roots of unity, is presented.
Given a $p$-adic field $K$ and a nilpotent uniform pro-$p$ group $G$, we prove that all primitive ideals in the $K$-rational Iwasawa algebra $KG$ are maximal, and can be reduced to a particular standard form. Setting $\mathcal{L}$ as the associated $\mathbb{Z}_p$-Lie algebra of $G$, our approach is to study the action of $KG$ on a Dixmier module $\widehat{D(λ)}$ over the affinoid envelope $\widehat{U(\mathcal{L})}_K$, and to prove that all primitive ideals can be reduced to annihilators of modules of this form.
Coradically graded Hopf algebras of tame corepresentation type
本文通过链箭图方法，刻画了具有对偶Chevalley性质的有限维Hopf代数的余表示类型，并给出了温和余表示型的充要条件。
In this article character groups of Hopf algebras are studied from the perspective of infinite-dimensional Lie theory. For a graded and connected Hopf algebra we construct an infinite-dimensional Lie group structure on the character group with values in a locally convex algebra. This structure turns the character group into a Baker--Campbell--Hausdorff--Lie group which is regular in the sense of Milnor. Furthermore, we show that certain subgroups associated to Hopf ideals become closed Lie subgroups of the character group. If the Hopf algebra is not graded, its character group will in general …
A noetherian Hopf algebra has bijective antipode
该研究证明，若任意域上的霍普夫代数的左或右正则模具有有限一致维数，则其具有双射反极，还解决了2006年斯克里亚宾提出的诺特霍普夫代数均具双射反极的猜想。
Let $p$ be an odd prime number and $K$ a number field having a primitive $p$-th root of unity $ζ.$ We prove that Nikshych's non-group theoretical Hopf algebra $H_p$, which is defined over $\mathbb{Q}(ζ)$, admits a Hopf order over the ring of integers $\mathcal{O}_K$ if and only if there is an ideal $I$ of $\mathcal{O}_K$ such that $I^{2(p-1)} = (p)$. This condition does not hold in a cyclotomic field. Hence this gives an example of a semisimple Hopf algebra over a number field not admitting a Hopf order over any cyclotomic ring of integers. Moreover, we show that, when a Hopf order over $\math…
In this paper we investigate a family of algebras endowed with a suitable non-degenerate bilinear form that can be used to define two different notions of dual for a given right ideal. We apply our results to the classification of the right ideals and their duals in the cyclic group algebra, in the Taft algebra and in another example of Hopf algebra arising as bosonization.
On the Semi-Abelianness of Affine Group Schemes
本文证明了在域$k$上交换Hopf代数的范畴是co-semi-abelian的，并推导出仿射群$k$-方案的范畴是semi-abelian的。
Rational Normal Curves, Chip Firing and Free Resolutions
本文通过筹码点火博弈与 parcycle 图的组合关联研究有理正规曲线，为每个 Cohen-Macaulay 初始单项式理想显式构造 Gröbner 退化及其极小自由消解，并给出胞腔消解、Hilbert 级数公式和 Eagon-Northcott 复形的组合解释。
New directions in the study of prime ideals in rational, nilpotent Iwasawa algebras
本文探讨了幂零p-可值（紧p进李）群的完成群代数（伊瓦萨代数）素理想，验证了关于其标准形式的猜想，排除了类型C的例外情况。
In previous work, Eli Aljadeff and the first-named author attached an algebra B_H of rational fractions to each Hopf algebra H. The generalized Noether problem is the following: for which finite-dimensional Hopf algebra H is B_H the localization of a polynomial algebra? A positive answer to this question when H is the algebra of functions on a finite group implies a positive answer for the classical Noether problem for the group. We show that the generalized Noether problem has a positive answer for all pointed finite-dimensional Hopf algebras over a field of characteristic zero. We actually g…
The character ring \CGL of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra \Sym$ of symmetric functions. Here we study the character rings \CO and \CSp of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that \CO and \CSp also admit natural Hopf algebra structures that are isomorphic to that of \CGL, and hence to \Sym. The isomorphisms are determined explicitly, along with the specification of standard bases for \CO and \CSp analog…
Study of the quotient module of a finite-dimensional Hopf subalgebra pair in order to compute its depth yields a relative Maschke Theorem, in which semisimple extension is characterized as being separable, and is therefore an ordinary Frobenius extension. We study the core Hopf ideal of a Hopf subalgebra, noting that the length of the annihilator chain of tensor powers of the quotient module is linearly related to the depth, if the Hopf algebra is semisimple. A tensor categorical definition of depth is introduced, and a summary from this new point of view of previous results are included. It i…
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