Narrow Results

In this paper, we discuss the group consisting of some good automorphisms of representation ring of the non-pointed Hopf algebra $D(n)$, the quotient of the non-pointed prime Hopf algebras of GK-dimension one, which is generated by $x,y,z$ with the relations:

We define notions of pivotal and ribbon objects in a monoidal category. These constructions give pivotal or ribbon monoidal categories from a monoidal category which is not necessarily with duals. We apply this construction to the braided monoidal category of Yetter–Drinfeld modules over a Hopf algebra. This gives rise to the notion of ribbon Yetter–Drinfeld modules over a Hopf algebra, which form ribbon categories. This gives an invariant of tangles.

We prove a double coset formula for induced representations of compact Lie groups. We apply it to the representation rings of unitary and symplectic groups to obtain Hopf algebras. We also construct a Heisenberg algebra representation based on the restiction and induction of representations of unitary groups.

We prove that Nichols algebras of irreducible Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊_n supported by 𝕊_n are infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊_n supported by A to be finite dimensional.

We identify the quantum isometry groups of spectral triples built on the symmetric groups with length functions arising from the nearest-neighbor transpositions as generators. It turns out that they are isomorphic to certain "doubling" of the group algebras of the respective symmetric groups. We discuss the doubling procedure in the context of regular multiplier Hopf algebras. In the last section we study the dependence of the isometry group of S_n on the choice of generators in the case n = 3. We show that two different choices of generators lead to nonisomorphic quantum isometry groups which exhaust the list of noncommutative noncocommutative semisimple Hopf algebras of dimension 12. This provides noncommutative geometric interpretation of these Hopf algebras.

Realizing the possibility suggested by Hardouin [Iterative q-difference Galois theory, J. Reine Angew. Math.644 (2010) 101–144], we show that her own Picard–Vessiot (PV) theory for iterative q-difference rings is covered by the (consequently, more general) framework, settled by Amano and Masuoka [Picard–Vessiot extensions of artinian simple module algebras, J. Algebra285 (2005) 743–767], of artinian simple module algebras over a cocommutative pointed Hopf algebra. An essential point is to represent iterative q-difference modules over an iterative q-difference ring R, by modules over a certain cocommutative ×_R-bialgebra. Recall that the notion of ×_R-bialgebras was defined by Sweedler [Groups of simple algebras, Publ. Math. Inst. Hautes Études Sci.44 (1974) 79–189], as a generalization of bialgebras.

We generalize results on the connection between existence and uniqueness of integrals and representation theoretic properties for Hopf algebras and compact groups. For this, given a coalgebra C, we study analogues of the existence and uniqueness properties for the integral functor Hom^C(C, -), which generalizes the notion of integral in a Hopf algebra. We show that the coalgebra C is co-Frobenius if and only if dim(Hom^C(C, M)) = dim(M) for all finite dimensional right (left) comodules M. As applications, we give a few new categorical characterizations of co-Frobenius, quasi-co-Frobenius (QcF) coalgebras and semiperfect coalgebras, and re-derive classical results of Lin, Larson, Sweedler and Sullivan on Hopf algebras. We show that a coalgebra is QcF if and only if the category of left (right) comodules is Frobenius, generalizing results from finite dimensional algebras, and we show that a one-sided QcF coalgebra is two-sided semiperfect. We also construct a class of examples derived from quiver coalgebras to show that the results of the paper are the best possible. Finally, we examine the case of compact groups, and note that algebraic integrals can be interpreted as certain skew-invariant measure theoretic integrals on the group.

We present explicit examples of finite tensor categories that are C₂-graded extensions of the corepresentation category of certain finite-dimensional non-semisimple Hopf algebras.

For a finite-index ${\mathrm{\text{II}}}_1$ subfactor $N\subset M$, 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 begin by considering a system which gives the usual fermion algebra when the appropriate limiting case is considered. We study its Hopf algebra structure. Investigation of its representation forces us to impose additional constraints on the system such that by defining a central hermitian operator s in terms of the operators defining the system, we obtain s-deformed particle algebra. Finally, we find a formula for the addition of s values for the d = 2 case.

In this paper, using a Hopf-algebraic method, we construct deformed Poincaré SUSY algebra in terms of twisted (Hopf) algebra. By adapting this twist deformed super-Poincaré algebra as our fundamental symmetry, we can see the consistency between the algebra and non(anti)commutative relation among (super)coordinates and interpret that symmetry of non(anti)commutative QFT is in fact twisted one. The key point is validity of our new twist element that guarantees non(anti)commutativity of space. It is checked in this paper for case. We also comment on the possibility of noncommutative central charge coordinate. Finally, because our twist operation does not break the original algebra, we can claim that (twisted) SUSY is not broken in contrast to the string inspired SUSY in non(anti)commutative superspace.

Domain wall solitons are the simplest topological objects in field theories. The conventional translational symmetry in a field theory is the generator of a one-parameter family of domain wall solutions, and induces a massless moduli field which propagates along a domain wall. We study similar issues in braided noncommutative field theories possessing Hopf algebraic translational symmetries. As a concrete example, we discuss a domain wall soliton in the scalar braided noncommutative field theory in Lie-algebraic noncommutative spacetime, which has a Hopf algebraic translational symmetry. We construct explicitly a one-parameter family of solutions in perturbation of the noncommutativity parameter. We then find the massless moduli field which propagates on the domain wall soliton. This work is based on arXiv:0711.3059.

Two approaches to the tangent space of a noncommutative space whose coordinate algebra is the enveloping algebra of a Lie algebra are known: the Heisenberg double construction and the approach via deformed derivatives, usually defined by procedures involving orderings among noncommutative coordinates or equivalently involving realizations via formal differential operators. In an earlier work, we rephrased the deformed derivative approach introducing certain smash product algebra twisting a semicompleted Weyl algebra. We show here that the Heisenberg double in the Lie algebra case, is isomorphic to that product in a nontrivial way, involving a datum ϕ parametrizing the orderings or realizations in other approaches. This way, we show that the two different formalisms, used by different communities, for introducing the noncommutative phase space for the Lie algebra type noncommutative spaces are mathematically equivalent.

If A is an associative algebra, then we can define the adjoint Lie algebra $A^{(-)}$ and Jordan algebra $A^{(+)}$. It is easy to see that any associative Rota–Baxter (RB) operator on A induces a Lie and Jordan RB operator on $A^{(-)}$ and $A^{(+)}$, respectively. Are there Lie (Jordan) RB operators, which are not associative RB operators? In this paper, we explore these questions for the Sweedler algebra $H_4$, which is a 4-dimensional non-commutative Hopf algebra. More precisely, we describe the RB operators on the adjoint Lie algebra $H_4^{(-)}$.

We describe an algorithm to compute the Kuperberg invariant of a combed or framed 3–manifold, starting from a presentation of such a manifold in terms of branched standard spines.

We describe an algorithm to compute the Kuperberg invariant of a combed or framed 3–manifold, starting from a presentation of such a manifold in terms of branched standard spines.

We study relations between group cohomologies and 3-manifold invariants. We first give a combinatorial construction of 3-manifold invariants using weight systems. We give examples of weight systems arising from a particular group cohomology. In the second part we show that these invariants can be obtained in a functorial way.

This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this approach is to take pure virtual braids as primary. The generators of the pure virtual braid group are abstract solutions to the algebraic Yang–Baxter equation. This point of view illuminates representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter equation. In this categorical framework, the virtual braid group is a natural group associated with the structure of algebraic braiding. We then point out how the category SC is related to categories associated with quantum algebras and Hopf algebras and with quantum invariants of virtual links.

Any finite-dimensional Hopf algebra $H$ is Frobenius and the stable category of $H$-modules is triangulated monoidal. To $H$-comodule algebras we assign triangulated module-categories over the stable category of $H$-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable $H$, our construction could be a starting point in the program of categorifying quantum invariants of 3-manifolds.

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.