We study the differential and Riemannian geometry of algebras A endowed with an action of a triangular Hopf algebra H and noncommutativity compatible with the associated braiding. The modules of one forms and of braided derivations are modules in a compact closed category of H -equivariant A -bimodules, whose internal morphisms correspond to tensor fields. Vector fields and forms approaches to curvature and torsion are proven to be equivalent by extending the Cartan calculus to left (right) A -module (not necessarily A -bimodule) connections. The Cartan structure equations and the Bianchi identities are derived. Existence and uniqueness of the Levi-Civita connection for arbitrary pseudo-Riemannian metrics is proven via a Koszul formula. The general theory includes Drinfeld twists of commutative geometries and also cotriangular Hopf algebras. It is illustrated with the example of the tensor square of Sweedler Hopf algebra which becomes a noncommutative Einstein manifold via a non-central metric.

Combinatorial Hopf algebras from restriction species with preorder cuts

In this paper we define three different notions of tensor products for Leibniz bimodules. The ``natural" tensor product of Leibniz bimodules is not always a Leibniz bimodule. In order to fix this, we introduce the notion of a weak Leibniz bimodule and show that the ``natural" tensor product of weak bimodules is again a weak bimodule. Moreover, it turns out that weak Leibniz bimodules are modules over a cocommutative Hopf algebra canonically associated to the Leibniz algebra. Therefore, the category of all weak Leibniz bimodules is symmetric monoidal and the full subcategory of finite-dimensional weak Leibniz bimodules is rigid and pivotal. On the other hand, we introduce two truncated tensor products of Leibniz bimodules which are again Leibniz bimodules. These tensor products induce a non-associative multiplication on the Grothendieck group of the category of finite-dimensional Leibniz bimodules. In particular, we prove that in characteristic zero for a finite-dimensional solvable Leibniz algebra this Grothendieck ring is an alternative power-associative commutative Jordan ring, but for a finite-dimensional non-zero semi-simple Leibniz algebra it is neither alternative nor a Jordan ring. 45 pages

ABSTRACT Let H be the 16-dimensional nontrivial semisimple Hopf algebra H b : y appeared in Kashina’s work [25]. We obtain all simple Yetter-Drinfeld modules over H and then determine all finite-dimensional Nichols algebras satisfying B ( V ) ≅ ⊗ i ∈ I B ( V i ) , where V = ⊕ i ∈ I V i , each V i is a simple object in H H Y D . Finally, we describe some liftings of those B ( V ) over H.

Firstly, we define and study the notions of a smash product for actions of multiplier left Hopf algebras on algebras and of an integral on such smash products. Then we construct an analogue of Radford’s biproduct in the framework of multiplier left Hopf algebras under assumption of a multiplier left Hopf algebra having an anti-bialgebra homomorphic left antipode. Finally, we study a duality theorem for smash products of a left Hopf algebra of dimension n which is a special multiplier left Hopf algebra.

For each braided category \mathcal{C} , we show that, under mild hypotheses, there is an associated category of “half braided algebras” and their bimodules internal to \mathcal{C} which is not only monoidal but even braided and balanced. We use this in the case where \mathcal{C} is the category of modules over a ribbon Hopf algebra to interpret stated skeins as a TQFT, namely, a braided balanced functor from a category of cobordisms to this category of algebras and their bimodules. Although our construction works in full generality, we relate in the special case of finite-dimensional ribbon factorizable Hopf algebras the stated skein functor to the Kerler–Lyubashenko TQFT by interpreting the former as the “endomorphisms” of the latter.

Frobenius monoidal functors induced by Frobenius extensions of Hopf algebras

The Projective Class Rings of Drinfeld doubles of pointed rank one Hopf algebras

Representations of the Drinfeld doubles of pointed rank one Hopf algebras

Homological integrals for weak Hopf algebras

A bstract We propose weak Hopf symmetry as a general framework to explore (1+1)D topological phases that exhibit non-invertible symmetries. Inspired by the Symmetry Topological Field Theory (SymTFT) description of quantum phases with non-invertible symmetry, we construct a lattice model by introducing two distinct topological boundary conditions for a weak Hopf lattice gauge theory. One boundary encodes the topological symmetry information, while the other incorporates the non-topological dynamics. The resulting model is termed the cluster ladder model. We demonstrate that the cluster state model is a special case of this broader class of lattice models exhibiting weak Hopf symmetry H × Ĥ , where H is a weak Hopf algebra and Ĥ is its dual weak Hopf algebra. On a closed manifold, the symmetry reduces to Cocom( H ) × Cocom( Ĥ ), corresponding to the cocommutative subalgebras of H × Ĥ . An essential weak Hopf sub-symmetry is Cocom( H ) × Rep( H ), which, in the finite group case, reduces to the familiar symmetry G × Rep( G ). To exactly solve the lattice model, we introduce a weak Hopf tensor network. Furthermore, we demonstrate how to construct the lattice realization of an arbitrary fusion category symmetry $$ \mathcal{S} $$ S via combining Tannaka-Krein reconstruction or weak Hopf tube algebra and the cluster ladder model.

In this note, we show that a right or left Noetherian graded algebra is affine if and only if its degree-zero part is affine. This result implies that a Noetherian Hopf algebra, when graded as an algebra, is affine if its degree-zero component is either a commutative or a cocommutative Hopf subalgebra. We further establish that the braided Hopf algebra of any right or left Noetherian graded Hopf algebra is affine. Communicated by Eric Jespers

Abstract Dualities play a central role in the study of quantum spin chains, providing insight into the structure of quantum phase diagrams and phase transitions. In this work, we study categorical dualities, which are defined as bounded-spread isomorphisms between algebras of symmetry-respecting local operators on a spin chain. We consider generalized global symmetries that correspond to unitary fusion categories, which are represented by matrix-product operator algebras. A fundamental question about dualities is whether they can be extended to quantum cellular automata on the larger algebra generated by all local operators in the the unit matrix-product operator sector. For on-site representations of Hopf algebra symmetries, this larger algebra is the usual tensor product quasi-local algebra. We present a solution to the extension problem using the machinery of Doplicher–Haag–Roberts bimodules. Our solution provides a crisp categorical criterion for when an extension of a duality exists. We show that the set of possible extensions form a torsor over the invertible objects in the relevant symmetry category. As a corollary, we obtain a classification result concerning dualities in the group case.

Abstract We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is based on defining 2D coproducts through horizontal and vertical maps that satisfy compatibility and associativity conditions, enabling the consistent growth of vector spaces over lattice sites. We present several examples of 2D bialgebras, including group-like and Lie algebra-inspired constructions and a quasi-1D coproduct instance that is applicable to Taft-Hopf algebras and to quantum groups. The approach is further applied to the quantum group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">[</mml:mo> <mml:mi>s</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:math> , for which we construct 2D generalizations of its generators, analyze q -deformed singlet states, and derive a 2D R-matrix satisfying an intertwining relation in the semiclassical limit. Additionally, we show how tensor network states, particularly projected entangled pair states, naturally induce 2D coalgebra structures when supplemented with appropriate boundary conditions. Our results establish a local and algebraically consistent method to embed quantum group symmetries into higher-dimensional lattice systems, potentially connecting to the emerging theory of fusion 2-categories and categorical symmetries in quantum many-body physics.

Infinitesimal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:mi mathvariant="script">R</mml:mi> </mml:math> -matrices for some families of Hopf algebras

Some Properties of the Redei–Berge Function and Related Combinatorial Hopf Algebras

For a given group [Formula: see text], we construct an invariant of flat [Formula: see text]-connections on 4-manifolds from a finite-type involutory quasitriangular Hopf [Formula: see text]-algebra. Hopf [Formula: see text]-algebras are generalizations of Hopf algebras, equipped with gradings by [Formula: see text]. In our construction, we color the dotted components of a Kirby diagram with elements of [Formula: see text] and employ the Hennings-type procedure. When [Formula: see text] is finite, we also define an invariant of 4-manifolds by summing the invariants over all flat [Formula: see text]-connections.

Abstract We consider quantum group representations Rep ⁡ ( G q ) \operatorname{Rep}(G_{q}) for a semisimple algebraic group 𝐺 at a complex root of unity 𝑞. Here we allow 𝑞 to be of any order. We first show that the Tannakian center in Rep ⁡ ( G q ) \operatorname{Rep}(G_{q}) is calculated via a twisting of Lusztig’s quantum Frobenius functor Rep ⁡ ( G ̌ ) → Rep ⁡ ( G q ) \operatorname{Rep}(\check{G})\to\operatorname{Rep}(G_{q}) , where G ̌ \check{G} is a dual group to 𝐺. We then consider the associated fiber category Vect ⊗ Rep ⁡ ( G ̌ ) Rep ⁡ ( G q ) \mathrm{Vect}\otimes_{\operatorname{Rep}{(\check{G})}}\operatorname{Rep}(G_{q}) over B ⁢ G ̌ B\check{G} , and show that this fiber is a finite, integral braided tensor category. Furthermore, when 𝐺 is simply connected and 𝑞 is of even order, the fiber in question is shown to be a modular tensor category. Finally, we exhibit a finite-dimensional quasitriangular quasi-Hopf algebra (also known as small quantum group) whose representations recover the tensor category Vect ⊗ Rep ⁡ ( G ̌ ) Rep ⁡ ( G q ) \mathrm{Vect}\otimes_{\operatorname{Rep}{(\check{G})}}\operatorname{Rep}(G_{q}) , and we describe the representation theory of this algebra in detail. At particular pairings of 𝐺 and 𝑞, our quasi-Hopf algebra is identified with Lusztig’s original finite-dimensional Hopf algebra from the ’90s. This work completes the author’s project from [C. Negron, Log-modular quantum groups at even roots of unity and the quantum Frobenius I, Comm. Math. Phys. 382 (2021), 2, 773–814].

We construct knot invariants from solutions to the Yang–Baxter equation associated to appropriately generalized left/right Yetter–Drinfel’d modules over a braided Hopf algebra with an automorphism. When applied to Nichols algebras, our method reproduces known knot polynomials and naturally produces multivariable polynomial invariants of knots. We analyze in detail the Nichols algebra of rank 1 , from which we recover the ADO and the colored Jones polynomials of a knot, and a Nichols algebra of rank 2 from which we obtain two sequences of knot invariants. One sequence starts with the product of two Alexander polynomials, and continues conjecturally with the Harper polynomial. The second sequence starts with the Links–Gould invariant (conjecturally), and then continues with a new 2-variable knot polynomial that detects chirality and mutation, and whose degree gives sharp bounds for the genus for a sample of 30 computed knots.

For a smooth affine algebraic group [Formula: see text], one can attach various D-module categories to it that admit convolution monoidal structure. We consider the derived category of D-modules on [Formula: see text], the stack [Formula: see text] and the category of Harish-Chandra bimodules. Combining the work of Beilinson–Drinfeld on D-modules and Hecke patterns with the recent work of the author with Dimofte and Py, we show that each of the above categories (more precisely the equivariant version) is monoidal equivalent to a localization of the DG category of modules of a graded Hopf algebra. As a consequence, we give an explicit braided monoidal structure to the derived category of D-modules on [Formula: see text], which when restricted to the heart, recovers the braiding of Bezrukavnikov–Finkelberg–Ostrik.