Related papers: Distinguished Pre-Nichols algebras

In this paper, we compute the dimension of the Hochschild cohomology groups of any $m$-cluster tilted algebra of type $\tilde{\mathbb{A}}$. Moreover, we give conditions on the bounded quiver of an $m$-cluster tilted algebra $\Lambda$ of…

We prove the Kazhdan-Lusztig correspondence for a class of vertex operator superalgebras which, via the work of Costello-Gaiotto, arise as boundary VOAs of topological B twist of 3d $\mathcal{N}=4$ abelian gauge theories. This means that we…

We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three…

In a previous work \cite{AS2} we showed how to attach to a pointed Hopf algebra A with coradical $\k\Gamma$, a braided strictly graded Hopf algebra R in the category $_{\Gamma}^{\Gamma}\Cal{YD}$ of Yetter-Drinfeld modules over $\Gamma$. In…

Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. They can also be considered as generalizations of ordinary root systems with rich structure and many new examples. On the other…

We give a systematic construction of Hopf algebra structures on braided cofree coalgebras. The relevant underlying structures are braided algebras and braided coalgebras. We provide some interesting examples of these algebras and coalgebras…

Let $(X, r)$ be any set-theoretical non-degenerate solution of the Yang-Baxter equation and $(X, \tilde r)$ be the derived solution of $(X, r)$. As for any braided vector space $(W_{X, r}, c)$ associated to $(X, r)$, is it possible to find…

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…

A pre-subgroup of a multiplicative unitary $V$ on a finite dimensionnal Hilbert space $H$ is a vector line $L$ in $H$ such that $V(L\otimes L)=L\otimes L$. We show that there are finitely many pre-subgroups, give a Lagrange theorem and…

Let $\mathcal{B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q} \in \mathbf{k}^{\theta \times \theta}$, where $\mathbf{k}$ is an algebraically closed field of characteristic…

In this paper, we explore a canonical connection between the algebra of $q$-difference operators $\widetilde{V}_{q}$, affine Lie algebra and affine vertex algebras associated to certain subalgebra $\mathcal{A}$ of the Lie algebra…

In this paper we construct "structural" pre-braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these pre-braidings seem original. On the other hand,…

We introduce the notion of a braided algebra and study some examples of these. In particular, R-symmetric and R-skew-symmetric algebras of a linear space V equipped with a skew-invertible Hecke symmetry R are braided algebras. We prove the…

Let S(n) be the category of invariant subspaces of nilpotent operators with nilpotency index at most n. Such submodule categories have been studied already in 1936 by Birkhoff, they have attracted a lot of attention in recent years, for…

This is a report on the present state of the problem of determining the dimension of the Nichols algebra associated to a rack and a cocycle. This is relevant for the classification of finite-dimensional complex pointed Hopf algebras whose…

Bialgebras and Hopf (bi)modules are typical algebraic structures with several interacting operations. Their structural and homological study is therefore quite involved. We develop the machinery of braided systems, tailored for handling…

Assume that $V$ is a braided vector space with diagonal type. It is shown that a monomial belongs to Nichols braided Lie algebra $\mathfrak L(V)$ if and only if this monomial is connected. A basis of Nichols braided Lie algebra and…

We show that the exterior algebra of a vector space $V$ of dimension greater than one admits a one-parameter family of braided Hopf algebra structures, arising from its identification with a Nichols algebra. We explicitly compute the…

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra K_{n}, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe…

To generalize some fundamental results on group schemes to the super context, we study the quotient sheaf $G \tilde{/} H$ of an algebraic supergroup $G$ by its closed supersubgroup $H$, in arbitrary characteristic $\neq$ 2. Our main theorem…