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This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics. [ABSTRACT FROM AUTHOR]

The theory of groups has a twofold symmetry, sending a group to its opposite. Groups invariant under the symmetry are abelian. The theory of quasigroups has a richer, sixfold symmetry, obtained by permuting the multiplication with its two divisions. The Sixfold Way identifies the various classes of quasigroups which are invariant under the respective subgroups of the symmetry group of the theory. Quantum quasigroups provide a self-dual framework to unify the study of quasigroups and Hopf algebras. The goal of this paper is to classify the symmetry classes of quantum quasigroups. Corresponding to the Sixfold Way for classical quasigroups, we are able to identify a Sevenfold Way for general classes exhibiting a symmetry, and initiate a study of a fuller symmetry which holds for linear quantum quasigroups. [ABSTRACT FROM AUTHOR]

We apply quantum group methods to quantum computing, starting with the notion of interacting Frobenius Hopf algebras for ZX calculus with noncommutative algebra and noncocommutative coalgebra. We introduce the notion of *-structures in ZX calculus at this algebraic level and construct examples based on the quantum group u q (sl 2) at a root of unity. We provide an abstract formulation of the Hadamard gate related to Hopf algebra self-duality. We then solve the problem of extending the notion of interacting Hopf algebras and ZX calculus to take place in a braided tensor category. In the ribbon case, the Hadamard gate coming from braided self-duality obeys a modular identity. We give the example of b q (sl 2), the self-dual braided version of u q (sl 2). [ABSTRACT FROM AUTHOR]

We introduce a method for construction of a covariant differential calculus over a Hopf algebra $A$ from a quantized calculus $da=[D,a]$, $a\in A$, where $D$ is a candidate for a Dirac operator for $A$. We recover the method of construction of a bicovariant differential calculus given by T. Brzeziński and S. Majid created from a central element of the dual Hopf algebra $A^\circ$. We apply this method to the Dirac operator for the quantum ${\rm SL(2)$ given by S. Majid. We find that the differential calculus obtained by our method is the standard bicovariant 4D-calculus. We also apply this method to the Dirac operator for the quantum $\rm SL(2)$ given by P. N. Bibikov and P. P. Kulish and show that the resulted differential calculus is $8$-dimensional.}

For a finite-index II 1 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. [ABSTRACT FROM AUTHOR]

In this study we introduce q-deformed split quaternions, that is, this deformation reduces to classical split quaternions as q-1 where q is a real parameter. It is also shown that there is a quantum group associated with q-deformed split quaternions, which is isomorphic to SUq (1,1). [ABSTRACT FROM AUTHOR]

We introduce a method for construction of a covariant differential calculus over a Hopf algebra A from a quantized calculus da = [D, a], a ∈ A, where D is a candidate for a Dirac operator for A. We recover the method of construction of a bicovariant differential calculus given by T. Brzeziński and S.Majid created from a central element of the dual Hopf algebra A°. We apply this method to the Dirac operator for the quantum SL(2) given by S.Majid. We find that the differential calculus obtained by our method is the standard bicovariant 4D-calculus. We also apply this method to the Dirac operator for the quantum SL(2) given by P.N.Bibikov and P.P.Kulish and show that the resulted differential calculus is 8-dimensional. [ABSTRACT FROM AUTHOR]

It is known that irreducible noncommutative differential structures over F p [ x ] are classified by irreducible monics m. We show that the cohomology H dR 0 (F p [ x ] ; m) = F p [ g d ] if and only if Trace(m)≠ 0, where g d = x p d − x and d is the degree of m. This implies that there are p − 1 pd ∑ k | d , p ∤ k μ M (k) p d k such noncommutative differential structures (μM the Möbius function). Motivated by killing this zero'th cohomology, we consider the directed system of finite-dimensional Hopf algebras A d = F p [ x ] / (g d) as well as their inherited bicovariant differential calculi Ω(Ad;m). We show that Ad = Cd ⊗χA1 is a cocycle extension where C d = A d ψ is the subalgebra of elements fixed under ψ(x) = x + 1. We also have a Frobenius-fixed subalgebra Bd of dimension 1 d ∑ k | d ϕ (k) p d k (ϕ the Euler totient function), generalising Boolean algebras when p = 2. As special cases, A 1 ≅ F p (ℤ / pℤ) , the algebra of functions on the finite group ℤ / pℤ , and we show dually that F p ℤ / pℤ ≅ F p [ L ] / (L p) for a 'Lie algebra' generator L with eL group-like, using a truncated exponential. By contrast, A2 over F 2 is a cocycle modification of F 2 ((ℤ / 2 ℤ) 2) and is a 1-dimensional extension of the Boolean algebra on 3 elements. In both cases we compute the Fourier theory, the invariant metrics and the Levi-Civita connections within bimodule noncommutative geometry. [ABSTRACT FROM AUTHOR]

We prove that the q-deformed unitary group, i.e., U q (N) , is the universal compact quantum group in the category of (compact) quantum groups which coact on the q-deformed odd sphere S q 2 N − 1 leaving the space spanned by the natural set of generators invariant and preserving the unique S U q (N) invariant functional on S q 2 N − 1. Using this, we identify U q (N) as the quantum group of orientation preserving isometries (in the sense of Bhowmick and Goswami [5]) for a natural spectral triple associated with S q 2 N − 1 constructed by Chakraborty and Pal [9]. [ABSTRACT FROM AUTHOR]

Let k be a field and let H denote a pointed Hopf k-algebra with antipode S. We are interested in determining the order of S. Building on the work done by Taft and Wilson in [7], we define an invariant for H, denoted mH, and prove that the value of this invariant is connected to the order of S. In the case where char k = 0, it is shown that if S has finite order then it is either the identity or has order 2 mH. If in addition H is assumed to be coradically graded, it is shown that the order of S is finite if and only if mH is finite. We also consider the case where char k = p > 0, generalizing the results of [7] to the infinite-dimensional setting. [ABSTRACT FROM AUTHOR]

Abstract We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra B in the category of comodules of a coquasitriangular Hopf algebra A has an associated coquasitriangular Hopf algebra Image 1. As an application we find new generators for c q [ S L 2 ] reduced at q a primitive odd root of unity with the remarkable property that their monomials are essentially a dual basis to the standard PBW basis of the reduced Drinfeld–Jimbo quantum enveloping algebra u q (s l 2). Our methods apply in principle for general c q [ G ] as we demonstrate for c q [ S L 3 ] at certain odd roots of unity. [ABSTRACT FROM AUTHOR]

Quantum quasigroups and quantum loops are self-dual objects providing a general framework for the nonassociative extension of quantum group techniques. This paper examines their one-sided analogues, which are not self-dual. Just as quantum quasigroups are the “quantum” version of quasigroups, so one-sided quantum quasigroups are the “quantum” version of left or right quasigroups.

We study the connections between one-sided Hopf algebras and one-sided quantum quasigroups, tracking the four possible invertibility conditions for the left and right composite morphisms that combine comultiplications and multiplications in these structures. The genuinely one-sided structures exhibit precisely two of the invertibilities, while it emerges that imposing one more condition often entails the validity of all four. A main result shows that under appropriate conditions, just one of the invertibility conditions is sufficient for the existence of a one-sided antipode. In the left Hopf algebra which is a variant of the quantum special linear group of two-dimensional matrices, it is shown explicitly that the right composite is not injective, and the left composite is not surjective. [ABSTRACT FROM AUTHOR]

Quantum quasigroups and quantum loops are self-dual objects providing a general framework for the nonassociative extension of quantum group techniques. This paper examines their one-sided analogues, which are not self-dual. Just as quantum quasigroups are the "quantum" version of quasigroups, so one-sided quantum quasigroups are the "quantum" version of left or right quasigroups. [ABSTRACT FROM AUTHOR]

We study super-braided Hopf algebras Λ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules Λ over a Hopf algebra A which are quotients of the augmentation ideal A as a crossed module by right multiplication and the adjoint coaction. Here super-bosonisation provides a bicovariant differential graded algebra on A. We introduce Λ providing the maximal prolongation, while the canonical braided-exterior algebra Λ = B (Λ) provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator ♯ by super-braided Fourier transform on B (Λ) and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on k[ S ] with its 3D calculus and obeying the q-Hecke relation ♯ = 1 + ( q − q )♯ in middle degree on k [ S L ] with its 4D calculus. Our work also provides a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras A whereby any subcoalgebra $\mathcal {L}\subseteq A$ defines a sub-braided Lie algebra and ${\Lambda }^{1}\subseteq \mathcal {L}^{*}$ provides the required data A → Λ. [ABSTRACT FROM AUTHOR]