arXiv subjects

N. Andruskiewitsch

Publications and source records attributed to N. Andruskiewitsch.

22 records · Page 2Linked to original sources

arXiv2001

Pointed Hopf algebras

This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras $A$ by first determining the graded Hopf algebra $\gr A$ associated to the coradical filtration of $A$. The $A_{0}$-coinvariants elements form a braided Hopf algebra $R$ in the category of Yetter-Drinfeld modules over the coradical $A_{0} = \ku Γ$, $Γ$ the group of group-like elements of $A$, and $\gr A \simeq R # A_{0}$. We call the braiding of the primitive elements of $R$ the infinitesimal braiding of $A$. If this braiding is of Cartan type \cite{AS2}, then it is often possible to determine $R$, to show that $R$ is generated as an algebra by its primitive elements and finally to compute all deformations or liftings, that is pointed Hopf algebras such that $\gr A \simeq R # \ku Γ$. In the last Chapter, as a concrete illustration of the method, we describe explicitly all finite-dimensional pointed Hopf algebras $A$ with abelian group of group-likes $G(A)$ and infinitesimal braiding of type $A_{n}$ (up to some exceptional cases). In other words, we compute all the liftings of type $A_n$; this result is our main new contribution in this paper.

math.QA↗

arXiv1998

Finite Dimensional Pointed Hopf Algebras with Abelian Coradical and Cartan matrices

In a previous work \cite{AS2} we showed how to attach to a pointed Hopf algebra A with coradical $\kΓ$, a braided strictly graded Hopf algebra R in the category $_Γ^Γ\Cal{YD}$ of Yetter-Drinfeld modules over $Γ$. In this paper, we consider a further invariant of A, namely the subalgebra R' of R generated by the space V of primitive elements. Algebras of this kind are known since the pioneering work of Nichols. It turns out that R' is completely determined by the braiding c:V\otimes V \to V \otimes V. We denote R' = B(V). We assume further that $Γ$ is finite abelian. Then c is given by a matrix (b_{ij}) whose entries are roots of unity; we also suppose that they have odd order. We introduce for these braidings the notion of "braiding of Cartan type" and we attach a generalized Cartan matrix to a braiding of Cartan type. We prove that B(V) is finite dimensional if its corresponding matrix is of finite Cartan type and give sufficient conditions for the converse statement. As a consequence, we obtain many new families of pointed Hopf algebras. When $Γ$ is a direct sum of copies of a group of prime order, the conditions hold and any matrix is of Cartan type. As a sample, we classify all the finite dimensional pointed Hopf algebras which are coradically graded, generated in degree one and whose coradical has odd prime dimension p. We also characterize coradically graded pointed Hopf algebras of order p^4, which are generated in degree one.

math.QA↗

arXiv1998

Braided Hopf algebras over non abelian finite groups

This is a survey of general aspects of the theory of braided Hopf algebras with emphasis on a special class of braided graded Hopf algebras named tobas. The interest on tobas arises from problems of classification of pointed Hopf algebras. We discuss tobas from different points of view following ideas of Lusztig, Nichols and Schauenburg. We then concentrate on braided Hopf algebras in the Yetter-Drinfeld category over H, where H is the group algebra of a non abelian finite group. We present some finite dimensional examples arising in an unpublished work by Milinski and Schneider.

math.QA↗

arXiv1998

Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p^3

We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra grad(A). Then grad(A) is a graded Hopf algebra, since the coradical A_0 of A is a Hopf subalgebra. In addition, there is a projection π: grad(A) \to A_0; let R be the algebra of coinvariants of π. Then, by a result of Radford and Majid, R is a braided Hopf algebra and grad(A) is the bosonization (or biproduct) of R and A_0: grad(A) is isomorphic to (R # A_0). The principle we propose to study A is first to study R, then to transfer the information to grad(A) via bosonization, and finally to lift to A. In this article, we apply this principle to the situation when R is the simplest braided Hopf algebra: a quantum linear space. As consequences of our technique, we obtain the classification of pointed Hopf algebras of order p^3 (p an odd prime) over an algebraically closed field of characteristic zero; with the same hypothesis, the characterization of the pointed Hopf algebras whose coradical is abelian and has index p or p^2; and an infinite family of pointed, non-isomorphic, Hopf algebras of the same dimension. This last result gives a negative answer to a conjecture of I. Kaplansky.

math.QA↗