Related papers: On Nichols bicharacter algebras

We prove {\rm (i)} Nichols algebra $\mathfrak B(V)$ of vector space $V$ is finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak L(V)$ is finite-dimensional; {\rm (ii)} If the rank of connected $V$ is $2$ and $\mathfrak…

Let $V$ be a braided vector space of diagonal type. Let $\mathfrak B(V)$, $\mathfrak L^-(V)$ and $\mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial…

We establish the relationship among Nichols algebras, Nichols braided Lie algebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra $\mathfrak B(V)$ is finite-dimensional if and only if Nichols braided Lie algebra…

It is shown that if $\mathfrak B(V) $ is connected Nichols algebra of diagonal type with $\dim V>1$, then $\dim (\mathfrak L^-(V)) = \infty$ $($resp. $ \dim (\mathfrak L(V)) = \infty $$)$ $($ resp. $ \dim (\mathfrak B(V)) = \infty $$)$ if…

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…

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…

Let ${\mathcal B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type with braiding matrix $\mathfrak{q}$, let $\mathcal{L}_{\mathfrak{q}}$ be the corresponding Lusztig algebra as in arXiv:1501.04518 and let…

Let $\mathbb K$ be an algebraically closed field of characteristic zero. Let $V$ be a module over the polynomial ring $\mathbb K[x,y]$. The actions of $x$ and $y$ determine linear operators $P$ and $Q$ on $V$ as a vector space over $\mathbb…

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…

Two algebras can be attached to a braided vector space $(V, c)$ in an intrinsic way; the FRT-bialgebra and the Nichols algebra $\toba(V, c)$. The FRT-bialgebra plays the r\^ole of the algebra of quantum matrices, whereas the r\^ole of the…

Let $\mathcal{B}_\mathfrak{q}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q}$. We consider the graded dual $\mathcal{L}_{\mathfrak{q}}$ of the distinguished pre-Nichols algebra…

A Lie algebra $L$ is said to be of breadth $k$ if the maximal dimension of the images of left multiplication by elements of the algebra is $k$. In this paper we give characterization of finite dimensional nilpotent Lie algebras of breadth…

We give a full classification of 6-dimensional nilpotent Lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic~2. To achieve the classification we use the action of the…

We introduce the notion of a braided Lie algebra consisting of a finite-dimensional vector space $\CL$ equipped with a bracket $[\ ,\ ]:\CL\tens\CL\to \CL$ and a Yang-Baxter operator $\Psi:\CL\tens\CL\to \CL\tens\CL$ obeying some axioms. We…

A double algebra is a linear space $V$ equipped with linear map $V\otimes V\to V\otimes V$. Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double…

We categorify the theory of Lie algebras beginning with a new notion of categorified vector space, or `2-vector space', which we define as an internal category in Vect, the category of vector spaces. We then define a `semistrict Lie…

A linear Lie rack structure on a finite dimensional vector space $V$ is a Lie rack operation $(x,y)\mapsto x\rhd y$ pointed at the origin and such that for any $x$, the left translation $\mathrm{L}_x:y\mapsto \mathrm{L}_x(y)= x\rhd y$ is…

We classify finite-dimensional pointed Hopf algebras with abelian coradical, up to isomorphism, and show that they are cocycle deformations of the associated graded Hopf algebra. More generally, for any braided vector space of diagonal type…

We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type {\rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({\rm YD} in short )modules over…

We formally define and study the distinguished pre-Nichols algebra $\widetilde{\mathcal{B}}(V)$ of a braided vector space of diagonal type $V$ with finite-dimensional Nichols algebra $\mathcal{B}(V)$. The algebra…