Narrow Results

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.

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 A, called $\mathrm{\Omega }$-algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study V-universal measuring coalgebras and V-universal comeasuring algebras between $\mathrm{\Omega }$-algebras A and B, relative to a fixed subspace V of $Vect(A,B)$. By considering the case $A=B$, we derive the notion of a V-universal (co)acting bialgebra (and Hopf algebra) for a given algebra A. 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 V-universal acting bi/Hopf algebra and the finite dual of the V-universal coacting bi/Hopf algebra under certain conditions on V in terms of the finite topology on $En{d}_F(A)$.

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 show that Nichols algebras of most simple Yetter–Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterion to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes in Chevalley and Steinberg groups.

We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor category. We also show that all objects in the tensor category have the structure of left-left crossed bimodules over the Hopf algebra object. For 4D factorizable topological quantum filed theories, we provide by analogous methods a construction of a Hopf algebra category.

Recently, in [25], Lie bialgebra structures on Lie algebras of generalized Weyl type were considered, which are shown to be triangular coboundary. In this paper, we quantize these algebras with their Lie bialgebra structures.

Generalising the shuffle product over a vector space, we define the sticky shuffle product over an associative algebra and endow it with a Hopf algebra structure. In various examples we exhibit connections with iterated sums, integrals and stochastic integrals and examine the significance of the Hopf algebra coproduct and antipode in these contexts. Finally we show that there is a homomorphism from the Yangian Hopf algebra Y(gl_N) to the corresponding sticky shuffle product Hopf algebra.

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.

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.

In this paper, we construct a bialgebraic and further a Hopf algebraic structure on top of subgraphs of a given graph. Further, we give the dual structure of this Hopf algebraic structure. We study the algebra morphisms induced by graph homomorphisms, and obtain a covariant functor from a graph category to an algebra category.

For $n,m\in \mathbb{N}$, let $H_{n,m}$ be the dual of the Radford algebra of dimension $n^{2}m$. We present new finite-dimensional Nichols algebras arising from the study of simple Yetter–Drinfeld modules over $H_{n,m}$. Along the way, we describe the simple objects in ${}_{H_{n,m}}^{H_{n,m}}{𝒴}{𝒟}$ and their projective envelopes. Then we determine those simple modules that give rise to finite-dimensional Nichols algebras for the case $n=2$. There are 18 possible cases. We present by generators and relations, the corresponding Nichols algebras on five of these eighteen cases. As an application, we characterize finite-dimensional Nichols algebras over indecomposable modules for $n=2=m$ and $n=2$, $m=3$, which recovers some results of the second and third author in the former case, and of Xiong in the latter.

Cualquier destino, por largo y complicado que sea, consta en realidad de un solo momento: el momento en que el hombre sabe para siempre quién es.

Jorge Luis Borges

In this paper, the authors develop an approach to construct a q-deformed Heisenberg-Virasoro algebra which is a Hom-Lie algebra, and investigate its central extensions and second cohomology group. Finally, quantum deformations of the Heisenberg-Virasoro algebra which provide a non-trivial Hopf structure are presented.

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.

Let A be a Hopf algebra over a field K of characteristic zero such that its coradical H is a finite-dimensional sub-Hopf algebra. Our main theorem shows that there is a gauge transformation ζ on A such that A^ζ ≅ Q#H where A^ζ is the dual quasi-bialgebra obtained from A by twisting its multiplication by ζ, Q is a connected dual quasi-bialgebra in and Q#H is a dual quasi-bialgebra called the bosonization of Q by H.

We give some applications of a Hopf algebra constructed from a group acting on another Hopf algebra $A$ as Hopf automorphisms, namely Molnar’s smash coproduct Hopf algebra. We find connections between the exponent and Frobenius–Schur indicators of a smash coproduct and the twisted exponents and twisted Frobenius–Schur indicators of the original Hopf algebra $A$. We study the category of modules of the smash coproduct.

We study the pre-Lie algebra of rooted trees $({𝒯},\to )$ and we define a pre-Lie structure on its doubling space $(V,⇝)$. Also, we find the enveloping algebras of the two pre-Lie algebras denoted, respectively, by $({{\mathscr{H}}}^{\prime },\star ,\mathrm{\Gamma })$ and $({{𝒟}}^{\prime },\star ,\chi )$. We prove that $({{𝒟}}^{\prime },\star ,\chi )$ is a module-bialgebra on $({{\mathscr{H}}}^{\prime },\star ,\mathrm{\Gamma })$ and we find some relations between the two pre-Lie structures.

In this paper, the author gives a complete set of simple Yetter–Drinfeld modules over the Suzuki algebra $A_{N2n+1}^{\mu \lambda }$ and investigates the Nichols algebras over those irreducible Yetter–Drinfeld modules. The finite-dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, Super type $A_2(q;{𝕀}_2)$ and the Nichols algebra $𝔲𝔣𝔬(8)$. And the involved finite-dimensional Nichols algebras of non-diagonal type are $12$, $4m$ and $m^2$-dimensional. The left three unsolved cases are set as open problems. In particular, all finite-dimensional Nichols algebras are given for simple Yetter–Dinfeld modules over $A_{13}^{+\lambda }$.

The aim of this work is to discuss the concepts of degeneration, deformation and rigidity of Hopf algebras and to apply them to the geometric study of the varieties of Hopf algebras. The main result is the description of the n-dimensional rigid Hopf algebras and the irreducible components for n<14 and n = p² with p a prime number.

For a finite tensor category $\mathcal{C}$, the action functor $\rho :\mathcal{C}\to \mathrm{Re}\text{x}\left(\mathcal{C}\right)$ is defined by ρ(X) = X ⨂ (−) for $X\in \mathcal{C}$, where $\mathrm{Re}\text{x}\left(\mathcal{C}\right)$ is the category of linear right exact endofunctors on $\mathcal{C}$. We show that ρ has a left and a right adjoint and demonstrate that adjoints of ρ are useful for dealing with certain (co)ends in $\mathcal{C}$. As an application, we give relations between some ring-theoretic notions and particular (co)ends.

A systematic definition and description of the basic properties of quantum simple Lie groups, quantum vector spaces, and quantum simple Lie algebras is given. A connection between these and the constructions of quantum deformations of the universal enveloping algebras of simple Lie algebras and the quantum double of Hopf algebras is established.