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V -Universal Hopf Algebras (co)Acting on Ω-Algebras
Abstract
We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced [A. L. Agore, A. S. Gordienko and J. Vercruysse, On equivalences of (co)module algebra structures over Hopf algebras, J. Noncommut. Geom., doi: 10.4171/JNCG/428.] bi/Hopf-algebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear maps between tensor powers of [Formula: see text], called [Formula: see text]-algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study [Formula: see text]-universal measuring coalgebras and [Formula: see text]-universal comeasuring algebras between [Formula: see text]-algebras [Formula: see text] and [Formula: see text], relative to a fixed subspace [Formula: see text] of [Formula: see text]. By considering the case [Formula: see text], we derive the notion of a [Formula: see text]-universal (co)acting bialgebra (and Hopf algebra) for a given algebra [Formula: see text]. In particular, this leads to a refinement of the existence conditions for the Manin–Tambara universal coacting bi/Hopf algebras. We establish an isomorphism between the [Formula: see text]-universal acting bi/Hopf algebra and the finite dual of the [Formula: see text]-universal coacting bi/Hopf algebra under certain conditions on [Formula: see text] in terms of the finite topology on [Formula: see text].
... Moreover, Sweedler showed the existence of a universal measuring coalgebra between any pair of algebras. This theory has been generalized in several ways, for example to the setting of closed monoidal categories in [9] and to more general types of algebraic structures, termed Ω-algebras in [1]. ...
... In the terminology of [3] this means that the category of algebras can be turned into a (klinear) semi-Hopf category. Following the work of [1], the same holds for any category of Ω-algebras. A natural question arises whether it is also possible to make this category into a Hopf-category, that is, to show the existence of an antipode for this semi-Hopf category. ...
... Dually to measuring coalgebras, one can consider comeasuring algebras. However, universal comeasurings only exist under suitable finiteness conditions (see [1]). Since Frobenius algebras are necessarily finite dimensional, universal comeasurings exist as well between any pair of Frobenius algebras. ...
We show that the universal measuring coalgebras between Frobenius algebras turn the category of Frobenius algebras into a Hopf category (in the sense of \cite{BCV}), and the universal comeasuring algebras between Frobenius algebras turn the category Frobenius algebras into a Hopf opcategory. We also discuss duality and compatibility results between these structures. Our theory vastly generalizes the well-known fact that any homomorphism beween Frobenius algebras is an isomorphism, but also allows to go beyond classical (iso)morphisms between Frobenius algebras, especially in finite characteristic, as we show by some explicit examples.
... Inspired by this, the notions of equivalence and universal Hopf algebras of (co)module structures on algebras were introduced in [3] as a natural generalization of the aforementioned universal group of a grading. This construction was unified in [4] with the universal (co)acting bi/Hopf algebras of Sweedler -Manin -Tambara, introducing the V -universal (co)acting bi/Hopf algebras for a given algebra A, where V is a unital subalgebra of End (A) and is the base field. The advantage of such a unified theory is that it simplifies the classification of (co)module structures (or even in certain cases makes it possible at all) by providing duality theorems. ...
... The advantage of such a unified theory is that it simplifies the classification of (co)module structures (or even in certain cases makes it possible at all) by providing duality theorems. Moreover, it is known that for a general infinite dimensional algebra A, the universal coacting bi/Hopf algebra of Manin -Tambara do not always exist (see [4,Section 4.5]) and the use of V provides the necessary restriction on the class of comodule structures under consideration to ensure the existence of the universal Hopf algebra for this class. Furthermore, following Sweedler [26], rather than considering just (universal) (co)actions of a bi/Hopf algebra on an algebra, one can also consider (universal) measurings of coalgebras between different algebras, similarly to considering homomorphisms between algebras rather than just endomorphisms of a single algebra. ...
... The main difference between existing literature and the approach we advocate here, is that all known general categorical results prove the existence of globally universal (i.e. universal among all measurings) (co)acting objects, and as already pointed out above, it is known [4,Section 4.5], that even in the case C = (Vect , ⊗), the category of vector spaces over a field , such global objects do not exist. Therefore, we propose to restrict the class of (co)measurings and (co)actions under consideration which allows for many new categories to be considered, including the ones previously mentioned and not covered by the existing literature. ...
The universal (co)acting bi/Hopf algebras introduced by Yu.\,I.~Manin, M.~Sweedler and D.~Tambara, the universal Hopf algebra of a given (co)module structure, as well as the universal group of a grading, introduced by J.~Patera and H.~Zassenhaus, find their applications in the classification of quantum symmetries. Typically, universal (co)acting objects are defined as initial or terminal in the corresponding categories and, as such, they do not always exist. In order to ensure their existence, we introduce the support of a given object, which generalizes the support of a grading and is used to restrict the class of objects under consideration. The existence problems for universal objects are formulated and studied in a purely categorical manner by seeing them as particular cases of the lifting problem for a locally initial object. We prove the existence of a lifting and, consequently, of the universal (co)acting objects under some assumptions on the base (braided or symmetric monoidal) category. In contrast to existing constructions, our approach is self-dual in the sense that we can use the same proof to obtain the existence of universal actions and coactions. In particular, when the base category is the category of vector spaces over a field, the category of sets or their duals, we recover known existence results for the aforementioned universal objects. The proposed approach allows us to apply our results not only to the classical categories of sets and vectors spaces and their duals but also to (co)modules over bi/Hopf algebras, differential graded vector spaces, G-sets and graded sets.
... The corresponding notions of equivalence and universal Hopf algebras of (co)module structures on algebras were introduced in [3], generalizing the aforementioned universal group of a grading. Furthermore, a unifying theory for universal Hopf algebras of (co)module structures and universal (co)acting bi/Hopf algebras of Sweedler -Manin -Tambara ( [21,17,22]) was introduced in [4], by considering V -universal (co)acting bi/Hopf algebras where V is a unital subalgebra of End (A) and is the base field. Motivation comes, on the one hand, from the fact that this unified theory simplifies and in certain cases even makes it possible at all to classify (co)module structures by means of duality results. ...
... Motivation comes, on the one hand, from the fact that this unified theory simplifies and in certain cases even makes it possible at all to classify (co)module structures by means of duality results. On the other hand, since the universal coacting bi/Hopf algebras of Manin -Tambara do not always exist [4,Section 4.5], V provides the necessary restriction on the class of comodule structures under consideration to ensure the existence of the universal Hopf algebra for this class. ...
... The first duality result for the (absolute) universal comeasuring algebra and measuring coalgebra, which generalizes the classical adjunction between the finite dual (of an algebra over a field) functor and the dual algebra (of a coalgebra over a field) functor (see e.g., [10,Theorem 1.5.22]), was established by D. Tambara in [22]. This duality was extended to V -universal (co)measuring (co)algebras and (co)acting bi/Hopf algebras over fields in [4,Theorems 3.20,4.14,4.15]. As it was proven in [4], the V -universal coacting Hopf algebras over fields admit a transparent description in terms of free algebras and relations, while the construction of V -universal acting Hopf algebras is not as explicit, since it involves subcoalgebras of cofree coalgebras. ...
In general, universal (co)measuring (co)monoids and universal (co)acting bi/Hopf monoids, which prove to be a useful tool in the classification of quantum symmetries, do not always exist. In order to ensure their existence, the support of a given object was recently introduced in \cite{AGV3} and used to restrict the class of objects considered when defining universal (co)acting objects. It is well-known that, in contrast with the universal coacting Hopf algebra, for actions on algebras over a field it is usually difficult to describe the universal acting Hopf algebra explicitly and this turns the duality theorem into an important investigation tool. In the present paper we establish duality results for universal (co)measuring (co)monoids and universal (co)acting bi/Hopf monoids in pre-rigid braided monoidal categories . In addition, when the base category is closed monoidal, we provide a convenient uniform approach to the aforementioned universal objects in terms of the cosupports, which in this case become subobjects of internal hom-objects. In order to explain our constructions, we use the language of locally initial objects. Known results from the literature are recovered when the base category is the category of vector spaces over a field. New cases where our results can be applied are explored, including categories of (co)modules over (co)quasitriangular Hopf algebras, Yetter-Drinfeld modules and dg-vector spaces.
... A particularly motivating example of a semi-Hopf category is the category of algebras, which is known to be enriched over coalgebras by means of Sweedler's universal measuring coalgebras, see [24], [26], [16], [4]. The results of our paper allow to consider the free or cofree Hopf category over this semi-Hopf category, which then leads then to a Hopf category structure on the category of algebras. ...
... Another case of particular interest, is given by the semi-Hopf category of algebras, which are known to be enriched over coalgebras by means of Sweedler's universal measuring coalgebras [4,24,26]. Our construction of the (co)free Hopf category over a semi-Hopf category allow to turn the category of algebras into a Hopf category. ...
... Indeed, in [18], A is supposed to be a graded algebra that is finite dimensional in each degree. This finiteness condition has been refined in [4]. In fact, following the constructions of [18] and [4], one would obtain a Hopf opcategory, rather then a Hopf category. ...
We show that under mild conditions on the monoidal base category , the category of Hopf -categories is locally presentable and deduce the existence of free and cofree Hopf categories. We also provide an explicit description of the free and cofree Hopf categories over a semi-Hopf category. One of the conditions on the base category , states that endofunctors obtained by tensoring with a fixed object preserve jointly monic families, which leads us to the notion of ``very flat monoidal product'', which we investigate in particular for module categories.
... The two results above remains valid if we take the category of Hopf algebras instead of bialgebras: in particular, the Hopf envelope of a(A), denoted by aut(A), is called in non-commutative geometry the non-commutative symmetry group of A [27] and its description is a very complicated matter. The existence and description of these universal (co)acting bialgebras/Hopf algebras has been considered recently in [2] in the context of Ω-algebras. The duality between Sweedler's and Manin-Tambara's objects has been extended to this general setting and necessary and sufficient conditions for the existence of the universal coacting bialgebras/Hopf algebras, which roughly explains the need for assuming finite-dimensionality in Manin-Tambara's constructions, are given. ...
... For an ndimensional Poisson algebra P , we denote P(P ) := P(P, P ) and we construct P(P ) as the quotient of the polynomial algebra k[X ij | i, j = 1, · · · , n] through an ideal generated by 2n 3 non-homogeneous polynomials of degree ≤ 2. P(P ) has a canonical bialgebra structure and Theorem 3.3 shows that P(P ) is the initial object of the category CoactBialg P of all commutative bialgebras coacting on P and, for this reason, we call it the universal coacting bialgebra of P . As in the case of Lie [5] or associative algebras [23], the universal bialgebra P(P ) has two important applications, which provide the theoretical answer for Poisson algebras, of the following open questions: (1) Describe explicitly the automorphisms group of a given Poisson algebra P ; (2) Describe and classify all G-gradings on P for a given abelian group G. More precisely, Theorem 3.6 proves that there exists an isomorphism of groups between the group of all Poisson automorphisms of P and the group of all invertible group-like elements of the finite dual P(P ) o . ...
... If B is a bialgebra, then G(B) is a monoid with respect to the multiplication on B. Throughout, for a bialgebra B, we denote by B o its finite dual. Recall that if H and L are two bialgebras then the abelian group Hom k (H, L) is an associative algebra under the convolution product [28]: (2) ), for all θ 1 , θ 2 ∈ Hom k (H, L) and h ∈ H. ...
We introduce the \emph{universal algebra} of two Poisson algebras P and Q as a commutative algebra satisfying a certain universal property. The universal algebra is shown to exist for any finite dimensional Poisson algebra P and several of its applications are highlighted. For any Poisson P-module U, we construct a functor from the category of A-modules to the category of Poisson Q-modules which has a left adjoint whenever U is finite dimensional. Similarly, if V is an A-module, then there exists another functor connecting the categories of Poisson representations of P and Q and the latter functor also admits a left adjoint if V is finite dimensional. If P is n-dimensional, then is the initial object in the category of all commutative bialgebras coacting on P. As an algebra, can be deescribed as the quotient of the polynomial algebra through an ideal generated by non-homogeneous polynomials of degree . Two applications are provided. The first one describes the automorphisms group as the group of all invertible group-like elements of the finite dual . Secondly, we show that for an abelian group G, all G-gradings on P can be explicitly described and classified in terms of the universal coacting bialgebra .
... The fact that c is a solution of the braid equation implies the very important fact that A(c) is a co-quasi-triangular bialgebra. That is, there exists a convolution-invertible bilinear map r : (1) , b (1) )a (2) (1) r(a (2) , b (2) ) This map is uniquely determined by CQT1-2-3 and ...
... The fact that c is a solution of the braid equation implies the very important fact that A(c) is a co-quasi-triangular bialgebra. That is, there exists a convolution-invertible bilinear map r : (1) , b (1) )a (2) (1) r(a (2) , b (2) ) This map is uniquely determined by CQT1-2-3 and ...
... The fact that c is a solution of the braid equation implies the very important fact that A(c) is a co-quasi-triangular bialgebra. That is, there exists a convolution-invertible bilinear map r : (1) , b (1) )a (2) (1) r(a (2) , b (2) ) This map is uniquely determined by CQT1-2-3 and ...
We prove that, in case A(c) = the FRT construction of a braided vector space (V,c) admits a weakly Frobenius algebra (e.g. if the braiding is rigid and its Nichols algebra is finite dimensional), then the Hopf envelope of A(c) is simply the localization of A(c) by a single element called the quantum determinant associated with the weakly Frobenius algebra. This generalizes a result of the author together with Gastón A. García in Farinati and García (J. Noncommutative Geom. 14(3), 879–911, 2020), where the same statement was proved, but with extra hypotheses that we now know were unnecessary. Along the way, we describe a concrete construction for a universal bialgebra associated to a finite dimensional vector space together V with some algebraic structure given by a family of maps . The Dubois-Violette and Launer Hopf algebra and the co-quasi triangular property of the FRT construction play a fundamental role in the proof.
... There is a current surge of interest in the study of universal quantum symmetries, see e.g., [1,2,5,8,9,10,11,18,22]. Notable results by Raedschelders and Van den Bergh in [18] showed that Manin's universal quantum groups of Koszul Artin-Schelter (AS) regular algebras with the same global dimensions have monoidally equivalent comodule categories. ...
We show that if two m-homogeneous algebras have Morita equivalent graded module categories, then they are quantum-symmetrically equivalent, that is, there is a monoidal equivalence between the categories of comodules for their associated universal quantum groups (in the sense of Manin) which sends one algebra to the other. As a consequence, any Zhang twist of an m-homogeneous algebra is a 2-cocycle twist by some 2-cocycle from its Manin's universal quantum group.
... We use Sweedler's notation with implied summation for both coalgebras (resp. bialgebras), as in ∆(c) = c (1) ⊗ c 2 , and for comodule structures: a right C-comodule structure ρ on a vector space V will be denoted by ρ(v) = v (0) ⊗ v (1) . When we need to be precise, the structures involved will be adorned. ...
Let and be two Lie algebras with finite dimensional and consider to be the corresponding universal algebra as introduced in \cite{am20}. Given an -module U and a Lie -module V we show that can be naturally endowed with a Lie -module structure. This gives rise to a functor between the category of Lie -modules and the category of Lie -modules and, respectively, to a functor between the category of -modules and the category of Lie -modules. Under some finite dimensionality assumptions, we prove that the two functors admit left adjoints which leads to the construction of universal -modules and universal Lie -modules as the representation theoretic counterparts of Manin-Tambara's universal coacting objects \cite{Manin, Tambara}.
... Автор надеется, что рукопись окажется полезной всем, кто решил познакомиться с данной темой на русском языке. От окончательного варианта диссертации эта работа отличается, в основном, наличием глав 2, 6 и 7, включающих в себя результаты статей [32,33,92,95], посвящённых эквивалентностям градуировок и (ко)модульных структур на алгебрах и связанным с ними категорным вопросам. ...
This manuscript is an extended version of the author's habilitation thesis defended on May 21, 2021 at M.V. Lomonosov Moscow State University. It is devoted to the study of (co)stability of radicals, existence of (co)invariant Levi and Wedderburn decompositions, structure of the corresponding simple algebras and codimension growth of polynomial identities in (co)module algebras over bi- and Hopf algebras and their generalizations. The main difference between this manuscript and the "official" thesis are additional chapters dealing with equivalences of gradings and Hopf algebra (co)actions, V-universal (co)acting bi- and Hopf algebras and related categorical questions. (In Russian.)
We show that if two m m -homogeneous algebras have equivalent graded module categories, then they are quantum-symmetrically equivalent, that is, there is a monoidal equivalence between the categories of comodules for their associated universal quantum groups (in the sense of Manin) which sends one algebra to the other. As a consequence, any Zhang twist of an m m -homogeneous algebra is a 2-cocycle twist by some 2-cocycle from its Manin’s universal quantum group.
The first historical encounter with Poisson-type algebras is with Hamiltonian mechanics. With the abstraction of many notions in Physics, Hamiltonian systems were geometrized into manifolds that model the set of all possible configurations of the system, and the cotangent bundle of this manifold describes its phase space, which is endowed with a Poisson structure. Poisson brackets led to other algebraic structures, and the notion of Poisson-type algebra arose, including transposed Poisson algebras, Novikov–Poisson algebras, or commutative pre-Lie algebras, for example. These types of algebras have long gained popularity in the scientific world and are not only of their own interest to study, but are also an important tool for researching other mathematical and physical objects.
We introduce the notion of quantum-symmetric equivalence of two connected graded algebras,
based on Morita–Takeuchi equivalences of their universal quantum groups, in the sense of Manin. We study
homological and algebraic invariants of quantum-symmetric equivalence classes, and prove that numerical
Tor-regularity, Castelnuovo–Mumford regularity, Artin–Schelter regularity, and the Frobenius property are
invariant under any Morita–Takeuchi equivalence. In particular, by combining our results with the work of
Raedschelders and Van den Bergh, we prove that Koszul Artin–Schelter regular algebras of a fixed global
dimension form a single quantum-symmetric equivalence class. Moreover, we characterize 2-cocycle twists
(which arise as a special case of quantum-symmetric equivalence) of Koszul duals, of superpotentials, of
superpotential algebras, of Nakayama automorphisms of twisted Frobenius algebras, and of Artin–Schelter
regular algebras. We also show that finite generation of Hochschild cohomology rings is preserved under certain 2-cocycle twists.
We construct a family of cogroupoids associated to preregular forms and recover the Morita–Takeuchi equivalence for Artin–Schelter regular algebras of dimension two, observed by Raedschelders and Van den Bergh. Moreover, we study the 2-cocycle twists of pivotal analogues of these cogroupoids, by developing a categorical description of preregularity in any tensor category that has a pivotal structure.
Let H be a Hopf algebra and be a 2-cocycle on H. Using the tensor equivalence between the comodule categories of H and of its 2-cocycle twist , we study twists of superpotentials, comodule algebras, and their associated universal quantum groups in the sense of Manin. In particular, we show the invariance of Artin--Schelter regularity of comodule algebras that are connected graded under a 2-cocycle twist of an infinite dimensional Hopf algebra. As a consequence, we show that Koszul AS-regular algebras of the same dimension and same Hilbert series are always 2-cocycle twists of each other, when viewed as comodule algebras over Manin's universal quantum groups.
For a finite-index II1 subfactor N ⊂ M, we prove the existence of a universal Hopf ∗-algebra (or, a discrete quantum group in the analytic language) acting on M in a trace-preserving fashion and fixing N pointwise. We call this Hopf ∗-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse.
We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) an equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra A , there exists a unique universal Hopf algebra H together with an H -(co)module structure on A such that any other equivalent (co)module algebra structure on A factors through the action of H . We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf–Galois extensions, actions of algebraic groups, and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. We apply support equivalence in the study of the asymptotic behavior of codimensions of H -identities and, in particular, to the analogue (formulated by Yu. A. Bahturin) of Amitsur's conjecture, which was originally concerned with ordinary polynomial identities. As an example, we prove this analogue for all unital H -module structures on the algebra F[x]/(x^2) of dual numbers.
Given a subfactor with finite index, we prove the existence of a universal Hopf -algebra (or, a discrete quantum group in the analytic language) which acts on M such that the action preserves the canonical trace of M and N is in the fixed point subalgebra for the action. We call this Hopf -algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably an arbitrary irreducible finite index depth two subfactor of type .
We study the existence of universal measuring comonoids P(A,B) for a pair
of monoids A, B in a braided monoidal closed category, and the associated
enrichment of a category of monoids over the monoidal category of comonoids. In
symmetric categories, we show that if A is a bimonoid and B is a
commutative monoid, then P(A,B) is a bimonoid; in addition, if A is a
cocommutative Hopf monoid then P(A,B) always is Hopf. If A is a Hopf
monoid, not necessarily cocommutative, then P(A,B) is Hopf if the fundamental
theorem of comodules holds; to prove this we give an alternative description of
the dualizable P(A,B)-comodules and use the theory of Hopf (co)monads. We
explore the examples of universal measuring comonoids in vector spaces, graded
spaces and differential graded spaces.
For any Koszul Artin-Schelter regular algebra A, we consider a version of the
universal Hopf algebra aut(A) coacting on A, introduced by Manin. To study the
representations (i.e. finite dimensional comodules) of this Hopf algebra, we
use the Tannaka-Krein formalism. Specifically, we construct an explicit
combinatorial rigid monoidal category U, equipped with a functor M to finite
dimensional vector spaces such that aut(A)= coend_U(M). Using this pair (U,M)
we show that aut(A) is quasi-hereditary as a coalgebra and in addition is
derived equivalent to the representation category of U.
We provide examples of non-surjective epimorphisms H→K in the category of Hopf algebras over a field, even with the additional requirement that K have bijective antipode, by showing that the universal map from a Hopf algebra to its enveloping Hopf algebra with bijective antipode is an epimorphism in HopfAlg, although it is known that it need not be surjective. Dual results are obtained for the problem of whether monomorphisms in the category of Hopf algebras are necessarily injective. We also notice that these are automatically examples of non-faithfully flat and respectively non-faithfully coflat maps of Hopf algebras.
We prove that both, the embedding of the category of Hopf algebras into that of bialgebras and the forgetful functor from the category of Hopf algebras to the category of algebras, have right adjoints; in other words: every bialgebra has a Hopf coreflection and on every algebra there exists a cofree Hopf algebra. In this way we give an affirmative answer to a forty years old problem posed by Sweedler. On the route the coequalizers and the coproducts in the category of Hopf algebras are explicitly described. Comment: to appear in Communications in Algebra
- A Elduque
- M V Kochetov