We develop a general theory of Hopf image of a Hopf algebra representation, with the associated concept of inner faithful representation, modelled on the notion of faithful representation of a discrete group. We study several examples, including group algebras, enveloping algebras of Lie algebras, pointed Hopf algebras, function algebras, twistings and cotwistings, and we present a Tannaka duality formulation of the notion of Hopf image.
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 $…
The Hopf actions on vertex operator algebras are investigated. If the action is semisimple, a Schur-Weyl type decomposition is obtained. When the Hopf algebra is finite dimensional and the action is faithful, the action is a group action. Moreover if the Hopf algebra is finite dimensional and the action is semisimple and inner faithful, the action is also a group action. In this case, inner faithfulness is equivalent to faithfulness.
We use a version of Haboush's theorem over complete local Noetherian rings to prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and separable (braided, symmetric) fusion categories from characteristic $p$ to characteristic zero (arXiv/math:0203060, Section 9), showing that, moreover, any isomorphism between such structures can be reduced modulo $p$. This fills a gap in arXiv/math:0203060, Subsection 9.3. We also show that lifting of semisimple cosemisimple Hopf algebras is a fully faithful functor, and prove that lifting induces an isomorphism on Picard and Brauer-Pic…
A family of algebra maps $H\to A_i$ whose common domain is a Hopf algebra is said to be jointly inner faithful if it does not factor simultaneously through a proper Hopf quotient of $H$. We show that tensor and free products of jointly inner faithful maps of Hopf algebras are again jointly inner faithful, generalizing a number of results in the literature on torus generation of compact quantum groups.
The notion of inner linear Hopf algebra is a generalization of the notion of discrete linear group. In this paper, we prove two general results that enable us to enlarge the class of Hopf algebras that are known to be inner linear: the first one is a characterization by using the Hopf dual, while the second one is a stability result under extensions. We also discuss the related notion of inner unitary Hopf *-algebra.
We construct a class of non-commutative, non-cocommutative, semisimple Hopf algebras of dimension $2n^2$ and present conditions to define an inner faithful action of these Hopf algebras on quantum polynomial algebras, providing, in this way, more examples of semisimple Hopf actions which do not factor through group actions. Also, under certain condition, we classify the inner faithful Hopf actions of the Kac-Paljutkin Hopf algebra of dimension $8$, $H_8$, on the quantum plane.
Let $R$ be an associative ring and $e,f$ idempotent elements of $R$. In this paper we introduce the notion of $(e,f)$-invertibility for an element of $R$ and use it to define inner actions of weak Hopf algebras. Given a weak Hopf algebra $H$ and an algebra $A$, we present sufficient conditions for $A$ to admit an inner action of $H$. We also prove that if $A$ is a left $H$-module algebra then $H$ acts innerly on the smash product $A\#H$ if and only if $H$ is a quantum commutative weak Hopf algebra.
We examine actions of finite-dimensional pointed Hopf algebras on central simple division algebras in characteristic 0. (By a Hopf action we mean a Hopf module algebra structure.) In all examples considered, we show that the given Hopf algebra does admit a faithful action on a central simple division algebra, and we construct such a division algebra. This is in contrast to earlier work of Etingof and Walton, in which it was shown that most pointed Hopf algebras do not admit faithful actions on fields. We consider all bosonizations of Nichols algebras of finite Cartan type, small quantum groups…
Ştefan spectral sequence for faithfully flat Hopf Galois extensions is multiplicative
School of Mathematics, Yangzhou University（扬州大学数学学院）；School of Mathematical Sciences, Laboratory of MEA (Ministry of Education), Shanghai Key Laboratory of PMMP, East China Normal University（华东师范大学数学科学学院教育部MEA实验室上海市PMMP重点实验室）；School of Mathematics, Shanghai University of Finance and Economics（上海财经大学数学学院）
This is a continuation of the authors' study of finite-dimensional pointed Hopf algebras H which act inner faithfully on commutative domains. As mentioned in Part I of this work, the study boils down to the case where H acts inner faithfully on a field. These Hopf algebras are referred to as Galois-theoretical. In this work, we provide classification results for finite-dimensional pointed Galois-theoretical Hopf algebras H of finite Cartan type. Namely, we determine when such H of type A_1^{\times r} and some H of rank two possess the Galois-theoretical property. Moreover, we provide necessary…
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.
We classify Hopf actions of Taft algebras T(n) on path algebras of quivers, in the setting where the quiver is loopless, finite, and Schurian. As a corollary, we see that every quiver admitting a faithful Z_n-action (by directed graph automorphisms) also admits inner faithful actions of a Taft algebra. Several examples for actions of the Sweedler algebra T(2) and for actions of T(3) are presented in detail. We then extend the results on Taft algebra actions on path algebras to actions of the Frobenius-Lusztig kernel u_q(sl_2), and to actions of the Drinfeld double of T(n).
This article is made up with two parts. In the first part, we generalize to the case when objects are faithfully flat over the ground ring, the full equivalence between the notions of Hopf-Galois extension and Hopf-Galois system. In the second part of the article, we show that the Kashiwara algebras introduced by Kashiwara in his study of crystal bases are Hopf-Galois objects.
We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if $π:A\to M_n(\mathbb C)$ is a finite dimensional representation of a Hopf $C^*$-algebra, we prove that the idempotent state associated to its Hopf image $A'$ must be the convolution Cesàro limit of the linear functional $ϕ=tr\circπ$. We discuss then some consequences of this result, notably to inner linearity questions.
Let H be a Hopf algebra with antipode S, and let A be an N-Koszul Artin-Schelter regular algebra. We study connections between the Nakayama automorphism of A and S^2 of H when H coacts on A inner-faithfully. Several applications pertaining to Hopf actions on Artin-Schelter regular algebras are given.
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 …
We study actions of semisimple Hopf algebras H on Weyl algebras A over a field of characteristic zero. We show that the action of H on A must factor through a group algebra; in other words, if H acts inner faithfully on A, then H is cocommutative. The techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.
We study actions of pointed Hopf algebras in the $\ZZ$-graded setting. Our main result classifies inner-faithful actions of generalized Taft algebras on quantum generalized Weyl algebras which respect the $\ZZ$-grading. We also show that generically the invariant rings of Taft actions on quantum generalized Weyl algebras are commutative Kleinian singularities.
The notion of Hopf center and Hopf cocenter of a Hopf algebra is investigated by the extension theory of Hopf algebras. We prove that each of them yields an exact sequence of Hopf algebras. Moreover the exact sequences are shown to satisfy the faithful (co)flatness condition. Hopf center and cocenter are computed for $\mathsf{U}_q(\mathfrak{g})$ and the Hopf algebra $\textrm{Pol}(\mathbb{G}_q)$, where $\mathbb{G}_q$ is the Drinfeld-Jimbo quantization of a compact semisimple simply connected Lie group $\mathbb{G}$ and $\mathfrak{g}$ is a simple complex Lie algebra.
We prove that commutative algebras in braided tensor categories do not admit faithful Hopf algebra actions unless they come from group actions. We also show that a group action allows us to see the algebra as the regular algebra in the representation category of the acting group.
Let A be a commutative unital algebra over an algebraically closed field k of characteristic not equal to 2, whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations. Let Q be a Hopf algebra that coacts on A inner-faithfully, while leaving V invariant. We prove that Q must be commutative when either: (i) the coaction preserves a non-degenerate bilinear form on V; or (ii) Q is co-semisimple, finite-dimensional, and char(k)=0.
We show that a Jordan-Hölder theorem holds for appropriately defined composition series of finite dimensional Hopf algebras. This answers an open question of N. Andruskiewitsch. In the course of our proof we establish analogues of the Noether isomorphism theorems of group theory for arbitrary Hopf algebras under certain faithful (co)flatness assumptions. As an application, we prove an analogue of Zassenhaus' butterfly lemma for finite dimensional Hopf algebras. We then use these results to show that a Jordan-Hölder theorem holds as well for lower and upper composition series, even though the f…
The link relation on simple subcoalgebras is used for decompositions of coalgebras. In this paper, we provide more sufficient conditions for this link relation, and prove a formula on the products between link-indecomposable components of Hopf algebras with the dual Chevalley property. Furthermore, we show that each of its component is generated by a simple subcoalgebra, as a faithfully flat module (in fact, a projective generator) over a Hopf subalgebra which is the component containing the unit element. Our conclusions generalize some relevant results on pointed Hopf algebras, which were est…
We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z_2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on cosemisimple Hopf algebras with a 2-dimensional irreducible comodule in the finite dimensional context. Finally, we give some applications to the classification of certain classes of semisimple Hopf alge…
In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin-Schelter regular algebras A of global dimension two, where A is a graded H-module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring A^H under such an action arises an analogue of a coordinate ring of a Kleinian singularity.
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.
Any finite-dimensional Hopf algebra has a left and a right integral. Conversely, Larsen and Sweedler showed that, if a finite-dimensional algebra with identity and a comultiplication with counit has a faithful left integral, it has to be a Hopf algebra. In this paper, we generalize this result to possibly infinite-dimensional algebras, with or without identity. We have to leave the setting of Hopf algebras and work with multiplier Hopf algebras. Moreover, whereas in the finite-dimensional case, there is a complete symmetry between the bialgebra and its dual, this is no longer the case in infin…
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.
We begin by considering the graded vector space with a basis consisting of rooted trees, graded by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices. We define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the multiplicities associated with each operator. The symmetric algebra on the vector space of rooted trees (after a deg…
Let $\mathbb{k}$ be an algebraically closed field of characteristic zero. Let $D$ be a division algebra of degree $d$ over its center $Z(D)$. Assume that $\mathbb{k}\subset Z(D)$. We show that a finite group $G$ faithfully grades $D$ if and only if $G$ contains a normal abelian subgroup of index dividing $d$. We also prove that if a finite dimensional Hopf algebra coacts on $D$ defining a Hopf-Galois extension, then its PI degree is at most $d^2$. Finally, we construct Hopf-Galois actions on division algebras of twisted group algebras attached to bijective cocycles.
In previous work, to each Hopf algebra H and each invertible right two-cocycle on H, Eli Aljadeff and the first-named author attached a subalgebra B of the free commutative Hopf algebra S generated by the coalgebra underlying H; the algebra B is the subalgebra of coinvariants of a generic Hopf Galois extension. In this paper we give conditions under which S is faithfully flat, or even free, as a B-module. We also show that B is generated as an algebra by certain elements arising from the theory of polynomial identities for comodule algebras developped jointly with Aljadeff.
The recent result of Brown and Zhang establishing Poincare duality in the Hochschild (co)homology of a large class of Hopf algebras is extended to right coideal subalgebras over which the Hopf algebra is faithfully flat, and applied to the standard Podles quantum 2-sphere.
We answer a question of A. Skalski and P.M. Sołtan (2016) about inner faithfulness of the S.~Curran's map of extending a quantum increasing sequence to a quantum permutation in full generality. To do so, we exploit some novel techniques introduced by Banica (2018) and Brannan, Chirvasitu, Freslon (2018) concerned with the Banica's conjecture regarding quantum permutation groups. Roughly speaking, we find a inductive setting in which the inner faithfulness of Curran's map can be boiled down to inner faithfulness of similar map for smaller algebras and then rely on inductive generation result fo…
A weakly equivariant Hopf algebra is a Hopf algebra A with an action of a finite group G up to inner automorphisms. We show that each weakly equivariant Hopf algebra can be replaced by a Morita equivalent algebra B with a strict action of G and with a coalgebra structure that leads to a tensor equivalent representation category. However, the coproduct of this strictification cannot, in general, be chosen to be unital, so that a strictification of the G-action can only be found on a weak Hopf algebra B.
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 …
We extend the universal differential calculus on an arbitrary Hopf algebra to a ``universal Cartan calculus''. This is accomplished by introducing inner derivations and Lie derivatives which act on the elements of the universal differential envelope. A new algebra is formulated by incorporating these new objects into the universal differential calculus together with consistent commutation relations. We also explain how to include nontrivial commutation relations into this formulation to obtain the ``generalized Cartan calculus''.
We extend the Larson-Sweedler theorem to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplike elements in the dual weak Hopf algebra. Defining distinguished left and right grouplike elements we derive the Radford formula for the fourth power of the antipode in a weak Hopf algebra and prove that the …
Motivated by a question of A.~Skalski and P.M.~Sołtan about inner faithfulness of the S.~Curran's map, we revisit the results and techniques of T.~Banica and J.~Bichon's Crelle paper and study some group-theoretic properties of the quantum permutation group on $4$ points. This enables us not only to answer the aforementioned question in positive in case $n=4, k=2$, but also to classify the automorphisms of $S_4^+$, describe all the embeddings $O_{-1}(2)\subset S_4^+$ and show that all the copies of $O_{-1}(2)$ inside $S_4^+$ are conjugate. We then use these results to show that the criterion w…
The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring $B$ can be embedded as a right coideal subalgebra into a Hopf algebra $A$ such that $A$ is faithfully flat as a $B$-module. In the present article such a Hopf algebra $A$ is constructed for the coordinate ring $B$ of the nodal cubic, thus further motivating the question which affine varieties are quantum homogeneous spaces.
Let (G,d) be a first order differential *-calculus on a *-algebra A. We say that a pair (π,F) of a *-representation πof A on a dense domain D of a Hilbert space and a symmetric operator F on D gives a commutator representation of G if there exists a linear mapping t:G -> L(D) such that t(adb)=π(a)i[F,π(b)], a,b in A. Among others, it is shown that each left-covariant *-calculus G of a compact quantum group Hopf *-algebra A has a faithful commutator representation. For a class of bicovariant *-calculi on A there is a commutator representation such that F is the image of a central element of the…
In this paper we establish a duality between etale Lie groupoids and a class of non-necessarily commutative algebras with a Hopf algebroid structure. For any etale Lie groupoid G over a manifold M, the groupoid algebra C_c(G) of smooth functions with compact support on G has a natural coalgebra structure over C_c(M) which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over C_c(M) we construct the associated spectral etale Lie groupoid G_sp(A) over M such that G_sp(C_c(G)) is naturally isomorphic to G. Both these constructions are functorial, and C_c is fully faithful left…
We define a noncommutative algebra of four basic objects within a differential calculus on quantum groups: functions, 1-forms, Lie derivatives and inner derivations, as the cross-product algebra associated with Woronowicz's (differential) algebra of functions and forms. This definition properly takes into account the Hopf algebra structure of the Woronowicz calculus. It also provides a direct proof of the Cartan identity.
The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the a…
We settle in this paper a question left open in our paper ``Modular Hecke algebras and their Hopf symmetry'', by showing how to extend the Rankin-Cohen brackets from modular forms to modular Hecke algebras. More generally, our procedure yields such brackets on any associative algebra endowed with an action of the Hopf algebra of transverse geometry in codimension one, such that the derivation corresponding to the Schwarzian derivative is inner. Moreover, we show in full generality that these Rankin-Cohen brackets give rise to associative deformations.
Let $U:\mathcal{C}\rightarrow\mathcal{D}$ be a strong monoidal functor between abelian monoidal categories admitting a right adjoint $R$, such that $R$ is exact, faithful and the adjunction $U\dashv R$ is coHopf. Building on the work of Balan, we show that $R$ is separable (resp., special) Frobenius monoidal if and only if $R(\mathbb{1}_{\mathcal{D}})$ is a separable (resp., special) Frobenius algebra in $\mathcal{C}$. If further, $\mathcal{C},\mathcal{D}$ are pivotal (resp., ribbon) categories and $U$ is a pivotal (resp., braided pivotal) functor, then $R$ is a pivotal (resp., ribbon) functor…
With the motivation of giving a more precise estimation of the quantum Brauer group of a Hopf algebra $H$ over a field $k$ we construct an exact sequence containing the quantum Brauer group of a Hopf algebra in a certain braided monoidal category. Let $B$ be a Hopf algebra in $\C=_H ^H\YD$, the category of Yetter-Drinfel'd modules over $H$. We consider the quantum Brauer group $\BQ(\C; B)$ of $B$ in $\C$, which is isomorphic to the usual quantum Brauer group $\BQ(k; B\rtimes H)$ of the Radford biproduct Hopf algebra $B\rtimes H$. We show that under certain symmetricity condition on the braidin…
Given a a Hopf algebra $H$, its left coideal subalgebra $A$ and a non-zero multiplicative functional $μ$ on $A$, we define the space of left $μ$-integrals $L^A_μ\subset A$. We observe that $\dim L^A_μ=1$ if $A$ is a Frobenius algebra and we conclude this equality for finite dimensional left coideal subalgebras of a weakly finite Hopf algebra. In general we prove that if $\dim L^A_μ>0$ then $\dim A <\infty$. Given a group-like element $g\in H$ we define the space $L^g_{ A}\subset A'$ of $g$-cointegrals on $ A$ and linking this concept with the theory of $μ$-integrals we observe that: - every se…
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum groups, we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions all into one big algebra. In particular we find a generalized Cartan identity that holds on the whole quantum universal enveloping algebra of the left-invariant vector fields and implicit commutation relations for a left-invariant basis of 1-forms.
We give an introduction to the theory of weak Hopf algebras proposed recently as a coassociative alternative of weak quasi-Hopf algebras. We follow an axiomatic approach keeping as close as possible to the "classical" theory of Hopf algebras. The emphasis is put on the new structure related to the presence of canonical subalgebras A^L and A^R in any weak Hopf algebra A that play the role of non-commutative numbers in many respects. A theory of integrals is developed in which we show how the algebraic properties of A, such as the Frobenius property, or semisimplicity, or innerness of the square…
