相关论文: On Finite dimensional Nichols algebras of diagonal…

It was conjectured in \texttt{\small arXiv:1606.02521} that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming that the rank is 2. We also show…

This is a contribution to the classification of finite-dimensional Hopf algebras over an algebraically closed field $\Bbbk$ of characteristic 0. Concretely, we show that a finite-dimensional Hopf algebra whose Hopf coradical is basic is a…

The classification of Nichols algebras is an essential step in the classification theory of pointed Hopf algebras by lifting method of N. Andruskiewitsch and H.-J. Schneider. Arithmetic root systems are invariants of Nichols algebras of…

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…

This paper contributes to the proof of the conjecture posed in arXiv:1606.02521, stating that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming…

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by lifting method of Andruskiewitsch and Schneider. Arithmetic root systems are invariants of…

Nichols algebras naturally appear in the classification of finite dimensional pointed Hopf algebras. Assuming only that the base field has characteristic zero several new finite dimensional rank 2 Nichols algebras of diagonal type are…

The paper purposes to contribute to the classification of pointed Hopf algebras by the method of Andruskiewitsch and Schneider. The structure of arithmetic root systems is enlightened such that their relation to ordinary root systems…

Let $H_8$ be the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra, which is also called Kac-Paljutkin algebra \cite{MR0208401}. All simple Yetter-Drinfel'd modules over $H_8$ are given. As for simple objects…

Let $(V,c)$ be a finite-dimensional braided vector space of diagonal type. We show that the Gelfand Kirillov dimension of the Nichols algebra $\mathfrak{B}(V)$ is finite if and only if the corresponding root system is finite, that is…

The Nichols algebras of diagonal type with finite root system are either of standard, super or (yet) unidentified type. A concrete description of the defining relations of all those Nichols algebras was given in \cite{A-exp presentation}.…

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, $\operatorname{GKdim}$ for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over $\mathbb…

We present new examples of finite-dimensional Nichols algebra over fields of characteristic 2 starting from braided vector spaces that are not of diagonal type, admit realizations as Yetter-Drinfeld modules over finite abelian groups and…

Nichols algebras are a fundamental building block of pointed Hopf algebras. Part of the classification program of finite-dimensional pointed Hopf algebras with the lifting method of Andruskiewitsch and Schneider is the determination of the…

We present algorithms to compute generalized root systems of Nichols algebras of diagonal type and of contragredient Lie superalgebras. As a consequence, we obtain an algorithm to compute the Lyndon words in the Kharchenko PBW basis…

We classify finite-dimensional Nichols algebras of Yetter-Drinfeld modules with indecomposable support over finite solvable groups in characteristic 0, using a variety of methods including reduction to positive characteristic. As a…

We prove that every pre-Nichols algebra of a nondiagonal object in the twisted Yetter-Drinfeld category ${_{\k G}^{\k G} {\mathcal{YD}^\Phi}}$ has infinite Gelfand-Kirillov dimension, where $G$ is a finite abelian group and $\Phi$ is a…

The concept of arithmetic root systems is introduced. It is shown that there is a one-to-one correspondence between arithmetic root systems and Nichols algebras of diagonal type having a finite set of (restricted) Poincare'-Birkhoff-Witt…

We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants such as real roots. The crucial ingredient is a `reflection' in the class of such Nichols algebras. We conclude the classifications of…

Between the braided vector spaces of diagonal type there exist some families whose associated Nichols algebras are infinite dimensional but with finite Gelfand-Kirillov dimension, called one-parameter families. We show that every…