Hopf Algebra Research Articles

Finite duals of affine prime regular Hopf algebras of GK-dimension one

<abstract><p>This paper is an attempt to construct a special kind of Hopf pairing $ \langle-, -\rangle:H^\bullet\otimes H\rightarrow {𝕜} $. Specifically, $ H^\bullet $ and $ H $ should be both affine, noetherian and of the same GK-dimension. In addition, some properties of them would be dual to each other. We test the ideas in two steps for all the affine prime regular Hopf algebras $ H $ of GK-dimension one: 1) We compute the finite duals $ H^\circ $ of them, which are given by generators and relations; 2) the Hopf pairings desired are determined by choosing certain Hopf subalgebras $ H^\bullet $ of $ H^\circ $, where $ \langle-, -\rangle $ becomes the evaluation.</p></abstract>

Some properties of the commutators of special linear quantum groups

This article is interested to a detailed computation of the commutators of the Hopf algebra Uq(sl(n)). It can betreated as a second way to computation the brackets of the Hopf algebra Uq(sl(n)) which could be introducing and understanding the Uq(sl(n)) for the researchers.

Graph polynomials associated with Dyson-Schwinger equations

Quantum motions are encoded by a particular family of recursive Hochschild equations in the renormalization Hopf algebra which represent Dyson-Schwinger equations, combinatorially. Feynman graphons, which topologically complete the space of Feynman diagrams of a gauge field theory, are considered to formulate some random graph representations for solutions of quantum motions. This framework leads us to explain the structures of Tutte and Kirchhoff-Symanzik polynomials associated with solutions of Dyson-Schwinger equations. These new graph polynomials are applied to formulate a new parametric representation for large Feynman diagrams and their corresponding Feynman rules.

4–Manifold invariants from Hopf algebras

The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaev's invariant. As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. We speculate that relaxing semisimplicity will lead to even richer invariants.

Irreducible representations of Hopf algebras over dihedral groups

Irreducible representations of Hopf algebras over dihedral groups

Hopf Differential Graded Galois Extensions

We introduce the concept of Hopf dg Galois extensions. For a finite dimensional semisimple Hopf algebra H and an H-module dg algebra R, we show that D(R#H)≅D(RH) is equivalent to that R/RH is a Hopf differential graded Galois extension. We present a weaker version of Hopf differential graded Galois extensions and show the relationships between Hopf differential graded Galois extensions and Hopf Galois extensions.

Solutions of the Yang–Baxter Equation Arising from Brauer Configuration Algebras

Currently, researching the Yang–Baxter equation (YBE) is a subject of great interest among scientists of diverse areas in mathematics and other sciences. One of the fundamental open problems is to find all of its solutions. The investigation deals with developing theories such as knot theory, Hopf algebras, quandles, Lie and Jordan (super) algebras, and quantum computing. One of the most successful techniques to obtain solutions of the YBE was given by Rump, who introduced an algebraic structure called the brace, which allows giving non-degenerate involutive set-theoretical solutions. This paper introduces Brauer configuration algebras, which, after appropriate specializations, give rise to braces associated with Thompson’s group F. The dimensions of these algebras and their centers are also given.

Amplitudes, Hopf algebras and the colour-kinematics duality

It was recently proposed that the kinematic algebra featuring in the colour-kinematics duality for scattering amplitudes in heavy-mass effective field theory (HEFT) and Yang-Mills theory is a quasi-shuffle Hopf algebra. The associated fusion product determines the structure of the Bern-Carrasco-Johansson (BCJ) numerators, which are manifestly gauge invariant and with poles corresponding to heavy-particle exchange. In this work we explore the deep connections between the quasi-shuffle algebra and general physical properties of the scattering amplitudes. First, after proving the double-copy form for gravitational HEFT amplitudes, we show that the coproducts of the kinematic algebra are in correspondence with factorisations of BCJ numerators on massive poles. We then study an extension of the standard quasi-shuffle Hopf algebra to a non-abelian version describing BCJ numerators with all possible gluon orderings. This is achieved by tensoring the original algebra with a particular Hopf algebra of orderings. In this extended version, a specific choice of the coproduct in the algebra of orderings leads to an antipode in the resulting Hopf algebra that has the interpretation of reversing the gluons’ order within each BCJ numerator.

Superpotentials and Quiver Algebras for Semisimple Hopf Actions

We consider the action of a semisimple Hopf algebra H on an m-Koszul Artin–Schelter regular algebra A. Such an algebra A is a derivation-quotient algebra for some twisted superpotential w, and we show that the homological determinant of the action of H on A can be easily calculated using w. Using this, we show that the smash product A#H is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra Λ to which A#H is Morita equivalent, generalising a result of Bocklandt–Schedler–Wemyss. We also show how Λ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan–Kirkman–Walton–Zhang follow using our techniques.

On the square of the antipode in a connected filtered Hopf algebra

It is well-known that the antipode $S$ of a commutative or cocommutative Hopf algebra satisfies $S^{2}=\operatorname*{id}$ (where $S^{2}=S\circ S$). Recently, similar results have been obtained by Aguiar, Lauve and Mahajan for connected graded Hopf algebras: Namely, if $H$ is a connected graded Hopf algebra with grading $H=\bigoplus_{n\geq0}H_n$, then each positive integer $n$ satisfies $\left( \operatorname*{id}-S^2\right)^n \left( H_n\right) =0$ and (even stronger) \[ \left( \left( \operatorname{id}+S\right) \circ\left( \operatorname{id}-S^2\right)^{n-1}\right) \left( H_n\right) = 0. \] For some specific $H$'s such as the Malvenuto--Reutenauer Hopf algebra $\operatorname{FQSym}$, the exponents can be lowered. In this note, we generalize these results in several directions: We replace the base field by a commutative ring, replace the Hopf algebra by a coalgebra (actually, a slightly more general object, with no coassociativity required), and replace both $\operatorname{id}$ and $S^2$ by "coalgebra homomorphisms" (of sorts). Specializing back to connected graded Hopf algebras, we show that the exponent $n$ in the identity $\left( \operatorname{id}-S^2\right) ^n \left( H_n\right) =0$ can be lowered to $n-1$ (for $n>1$) if and only if $\left( \operatorname{id} - S^2\right) \left( H_2\right) =0$. (A sufficient condition for this is that every pair of elements of $H_1$ commutes; this is satisfied, e.g., for $\operatorname{FQSym}$.)

Topological dualities in the Ising model

We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in $2$ dimensions, with electromagnetic duality for finite gauge theories in $3$ dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.

Finite-dimensional Nichols algebras over the Suzuki algebras I: simple Yetter-Drinfeld modules of $A_{N\,2n}^{\mu\lambda}$

The Suzuki algebra $A_{Nn}^{\mu \lambda}$, introduced by Suzuki Satoshi in 1998, is a class of cosemisimple Hopf algebras. It is not categorically Morita-equivalent to a group algebra in general. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra $A_{N\,2n}^{\mu\lambda}$ and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, $A_2\times A_2$, Super type ${\bf A}_{2}(q;\I_2)$ and the Nichols algebra $\ufo(8)$. There are $64$, $4m$ and $m^2$-dimensional Nichols algebras of non-diagonal type over $A_{N\,2n}^{\mu \lambda}$. The $64$-dimensional Nichols algebras are of dihedral rack type $\Bbb{D}_4$. The $4m$ and $m^2$-dimensional Nichols algebras $\mathfrak{B}(V_{abe})$ discovered first by Andruskiewitsch and Giraldi can be realized in the category of Yetter-Drinfeld modules over $A_{Nn}^{\mu \lambda}$. Using a result of Masuoka, we prove that $\dim\mathfrak{B}(V_{abe})=\infty$ under the condition $b^2=(ae)^{-1}$, $b\in\Bbb{G}_{m}$ for $m\geq 5$.

Representations of some associative pseudoalgebras

Abstract In this paper, we generalize Schur–Weyl duality and Morita Theorem on associative algebras to those on associative H-pseudoalgebras. Meanwhile, we get a plenty of associative H-pseudoalgebras over a cocommutative Hopf algebra H

Hopf Structures in the Representation Theory of Direct Products

Combinatorial Hopf algebras give a linear algebraic structure to infinite families of combinatorial objects, a technique further enriched by the categorification of these structures via the representation theory of families of algebras. This paper examines a fundamental construction in group theory, the direct product, and how it can be used to build representation theoretic Hopf algebras out of towers of groups. A key special case gives us the noncommutative symmetric functions NSym, but there are many things that we can say for the general Hopf algebras, including the structure of their character groups and a formula for the antipode.

Twisting of Graded Quantum Groups and Solutions to the Quantum Yang-Baxter Equation

AbstractLet H be a Hopf algebra that is $\mathbb Z$ ℤ -graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of H to be a Zhang twist of H. In particular, we introduce the notion of a twisting pair for H such that the Zhang twist of H by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin’s universal quantum groups associated with quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction.

Invertible skew pairings and crossed products for weak Hopf algebras

In this paper we work with invertible skew pairings for weak bialgebras in a symmetric monoidal category where every idempotent morphism splits. We prove that this kind of skew pairings induces examples of weak distributive laws and therefore they provide weak wreath products. Also we will show that they define weakly comonoidal mutually weak inverse pairs of weak distributive laws and, by the results proved by G. Böhm and J. Gómez-Torrecillas, we obtain weak wreath products that become weak bialgebras with respect to the tensor product coalgebra structure. As an application, we will show that the Drinfel'd double of a finite weak Hopf algebra can be constructed using the weak wreath product associated to an invertible $1$-skew pairing.

Non-commutative stochastic processes with independent increments

This paper is on the research of Wilhelm von Waldenfels in the mathematical field of quantum (or non-commutative) probability theory. Wilhelm von Waldenfels certainly was one of the pioneers of this field. His idea was to work with moments and to replace polynomials in commuting variables by free algebras which play the role of algebras of polynomials in non-commuting quantities. Before he contributed to quantum probability he already worked with free algebras and free Lie algebras. One can imagine that this helped to create his own special algebraic method which proved to be so very fruitful. He came from physics. His PhD thesis, supervised by Heinz König, was in probability theory, in the more modern and more algebraic branch of probability theory on groups. Maybe the three, physics, abstract algebra and probability, must have been the best prerequisites to become a pioneer, even one of the founders, of quantum probability. We concentrate on a small part of the scientific work of Wilhelm von Waldenfels. The aspects of physics are practically not mentioned at all. There is nothing on his results in classical probability on groups (Waldenfels operators). This is an attempt to show how the concepts of non-commutative notions of independence and of Lévy processes on structures like Hopf algebras developed from the ideas of Wilhelm von Waldenfels.

Structures of adjoint-stable algebras over factorizable Hopf algebras

For a quasi-triangular Hopf algebra (H, R), there is a notion of transmuted braided group HR of H introduced by Majid. The transmuted braided group HR is a Hopf algebra in the braided category . The R-adjoint-stable algebra associated with any simple left HR -comodule is defined by the authors, and is used to characterize the structure of all irreducible Yetter-Drinfeld modules in . In this note, we prove for a semisimple factorizable Hopf algebra (H, R) that any simple subcoalgebra of HR is H-stable and the R-adjoint-stable algebra for any simple left HR -comodule is anti-isomorphic to H. As an application, we characterize all irreducible Yetter-Drinfeld modules.

Tracelet Hopf Algebras and Decomposition Spaces (Extended Abstract)

Tracelets are the intrinsic carriers of causal information in categorical rewriting systems. In this work, we assemble tracelets into a symmetric monoidal decomposition space, inducing a cocommutative Hopf algebra of tracelets. This Hopf algebra captures important combinatorial and algebraic aspects of rewriting theory, and is motivated by applications of its representation theory to stochastic rewriting systems such as chemical reaction networks.

Faà di Bruno Hopf algebras

This is a short review on the Faà di Bruno formulas, implementing composition of real-analytic functions, and a Hopf algebra associated to such formulas. This structure provides, among several other things, a short proof of the Lie-Scheffers theorem, and relates the Lagrange inversion formulas with antipodes. It is also the maximal commutative Hopf subalgebra of the one used by Connes and Moscovici to study diffeomorphisms in a noncommutative geometry setting. The link of Faà di Bruno formulas with the theory of set partitions is developed in some detail.