Quantum affine algebras at small root of unity

We study the Frobenius-Lusztig kernel for quantum affine algebras at root of unity of small orders that are usually excluded in literature. These cases are somewhat degenerate and we find that the kernel is in fact mostly related to different affine Lie algebras, some even of larger rank, that exceptionally sit inside the quantum affine algebra. This continues the authors study for quantum groups associated to finite-dimensional Lie algebras in [Len14c].Comment: 43 page

New Large-Rank Nichols Algebras Over Nonabelian Groups With Commutator Subgroup Z_2

In this article, we explicitly construct new finite-dimensional, link-indecomposable Nichols algebras with Dynkin diagrams of type An,Cn,Dn,E6,E7,E8,F4 over any group G with commutator subgroup isomorphic to Z_2.The construction is generic in the sense that the type just depends on the rank and center of G, and thus positively answers for all groups of this class a question raised by Susan Montgomory in 1995 [Mont95][AS02]. Our construction uses the new notion of a covering Nichols algebra as a special case of a covering Hopf algebra [Len12] and produces non-faithful Nichols algebras. However, we give faithful examples of Doi twists for type A3,C3,D4,F4 over several nonabelian groups of order 16 and 32. These are hence the first known examples of faithful, finite-dimensional, link-indecomposable Nichols algebras of rank >2 over nonabelian groups.Comment: Updated some references. A shortend version has appeared in Journal of Algebr

Root Systems In Finite Symplectic Vector Spaces

We study subsets in possibly degenerate symplectic vector spaces over finite fields, which are stable under a given Coxeter/Weyl reflection group. These symplectic root systems provide crucial combinatorical data to classify finite-dimensional Nichols algebras for nilpotent groups G over the complex numbers [Len13a], where the symplectic form is given by the group's commutator map. For example, the degree of degeneracy of the symplectic root system determines the size of the center of G. In this article we classify symplectic root systems over the finite field F_2, where symplectic just means isotropic. We prove that every Dynkin diagram admits, up to symplectic isomorphisms, a unique minimal symplectic root system over F_2 and thus requires a specific degree of degeneracy of the symplectic vector space. Any non-minimal symplectic root system turns out to be a quotient of a minimal one by a universal property. As examples and for further applications we explicitly construct all symplectic root systems for Cartan matrices resp. Dynkin diagrams of type ADE.Comment: Final version to appear in Communications in Algebra. Includes a list with all symplectic root systems of type ADE over F_

A simplicial complex of Nichols algebras

We translate the concept of restriction of an arrangement in terms of Hopf algebras. In consequence, every Nichols algebra gives rise to a simplicial complex decorated by Nichols algebras with restricted root systems. As applications, some of these Nichols algebras provide Weyl groupoids which do not arise for Nichols algebras over finite groups and in fact we realize all root systems of finite Weyl groupoids of rank greater than three. Further, our result explains the root systems of the folded Nichols algebras over nonabelian groups and of generalized Satake diagrams.Comment: 48 pages, 27 figures, final version to appear in Mathematische Zeitschrif

Logarithmic conformal field theories of type Bn,ℓ=4B_n,\ell=4 and symplectic fermions

There are important conjectures about logarithmic conformal field theories (LCFT), which are constructed as kernel of screening operators acting on the vertex algebra of the rescaled root lattice of a finite-dimensional semisimple complex Lie algebra. In particular their representation theory should be equivalent to the representation theory of an associated small quantum group. This article solves the case of the rescaled root lattice $B_n/\sqrt{2}$ as a first working example beyond $A_1/\sqrt{p}$. We discuss the kernel of short screening operators, its representations and graded characters. Our main result is that this vertex algebra is isomorphic to a well-known example: The even part of $n$ pairs of symplectic fermions. In the screening operator approach this vertex algebra appears as an extension of the vertex algebra associated to rescaled $A_1^n$, which are $n$ copies of the even part of one pair. The new long screenings give the global $C_n$-symmetry. The extension is due to a degeneracy in this particular case: Rescaled long roots still have even integer norm. The associated quantum group of divided powers has similar degeneracies [Lent16]: It contains the small quantum group of type $A_1^n$ and the Lie algebra $C_n$. Recent results [FGR17b] on symplectic fermions suggest finally the conjectured category equivalence to this quantum group. We also study the other degenerate cases of a quantum group, giving extensions of LCFT's of type $D_n,D_4,A_2$ with larger global symmetry $B_n,F_4,G_2$.Comment: 41 page

A theorem on roots of unity and a combinatorial principle

Given a finite set of roots of unity, we show that all power sums are non-negative integers iff the set forms a group under multiplication. The main argument is purely combinatorial and states that for an arbitrary finite set system the non-negativity of certain alternating sums is equivalent to the set system being a filter. As an application we determine all discrete Fourier pairs of $\{0,1\}$-matrices. This technical result is an essential step in the classification of $R$-matrices of quantum groups.Comment: We have proven the more general combinatorial statement and made some other minor improvement

On monoidal autoequivalences of the category of Yetter-Drinfeld modules over a group: The lazy case

An interesting open question is to determine the group of monoidal autoequivalences of the category of Yetter-Drinfeld modules over a finite group $G$, or equivalently the group of Bigalois objects over the dual of the Drinfeld double $DG$. In particular one would hope to decompose this group into terms related to monoidal autoequivalences for the group algebra, the dual group algebra and interaction terms. We report on our progress in this question: We first prove a decomposition of the group of Hopf algebra automorphisms of the Drinfeld double into three subgroups, which reduces in the case $G=\mathbb{Z}_p^n$ to a Bruhat decomposition of $\mathrm{GL}_{2n}(\mathbb{Z}_p)$. Secondly, we propose a Kuenneth-like formula for the Hopf algebra cohomology of $DG^*$ into three terms and prove partial results in the case of lazy cohomology. We use these results for the calculation of the Brauer-Picard group in the lazy case in [LP15].Comment: 30 pages, v2: typos corrected. Automorphisms and (more extensive) cohomology calculations exported from arXiv:1506.0783

A decomposition of the Brauer-Picard group of the representation category of a finite group

We present an approach of calculating the group of braided autoequivalences of the category of representations of the Drinfeld double of a finite dimensional Hopf algebra $H$ and thus the Brauer-Picard group of $H$-$\mathrm{mod}$. We consider two natural subgroups and a subset as candidates for generators. In this article $H$ is the group algebra of a finite group $G$. As our main result we prove that any element of the Brauer-Picard group, fulfilling an additional cohomological condition, decomposes into an ordered product of our candidates. For elementary abelian groups $G$ our decomposition reduces to the Bruhat decomposition of the Brauer-Picard group, which is in this case a Lie group over a finite field. Our results are motivated by and have applications to symmetries and defects in $3d$-TQFT and group extensions of fusion categories.Comment: 37 pages, v2: Automorphisms and (more extensive) cohomology calculations exported to arXiv:1511.03871, other minor improvements, v3: typos corrected, other improvement

Nash Equilibria And Partition Functions Of Games With Many Dependent Players

We discuss and solve a model for a game with many players, where a subset of truely deciding players is embedded into a hierarchy of dependent agents. These interdependencies modify the game matrix and the Nash equilibria for the deciding players. In a concrete example, we recognize the partition function of the Ising model and for high dependency we observe a phase transition to a new Nash equilibrium, which is the Pareto-efficient outcome. An example we have in mind is the game theory for major shareholders in a stock market, where intermediate companies decide according to a majority vote of their owners and compete for the final profit. In our model, these interdependency eventually forces cooperation.Comment: 14 page

Partially dualized Hopf algebras have equivalent Yetter-Drinfel'd modules

Given a Hopf algebra $H$ and a projection $H\to A$ to a Hopf subalgebra, we construct a Hopf algebra $r(H)$, called the partial dualization of $H$, with a projection to the Hopf algebra dual to $A$. This construction provides powerful techniques in the general setting of braided monoidal categories. The construction comprises in particular the reflections of generalized quantum groups, arxiv:1111.4673 . We prove a braided equivalence between the Yetter-Drinfel'd modules over a Hopf algebra and its partial dualization