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Register a new userPrimitive Deformations of Quantum $p$-groups
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For finite-dimensional Hopf algebras, their classification in characteristic $0$ (e.g. over $mathbb{C}$) has been investigated for decades with many fruitful results, but their structures in positive characteristic have remained elusive. In this paper, working over an algebraically closed field $mathbf{k}$ of prime characteristic $p$, we introduce the concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional connected Hopf algebras which are almost primitively generated; that is, these connected Hopf algebras are $p^{n+1}$-dimensional, whose primitive spaces are abelian restricted Lie algebras of dimension $n$. We illustrate this technique for the case $n=2$. Together with our preceding results in arXiv:1309.0286, we provide a complete classification of $p^3$-dimensional connected Hopf algebras over $mathbf{k}$ of characteristic $p>2$.
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In general the endomorphisms of a non-abelian group do not form a ring under the operations of addition and composition of functions. Several papers have dealt with the ring of functions defined on a group which are endomorphisms when restricted to the elements of a cover of the group by abelian subgroups. We give an algorithm which allows us to determine the elements of the ring of functions of a finite $p$-group which arises in this manner when the elements of the cover are required to be either cyclic or elementary abelian of rank $2$. This enables us to determine the actual structure of such a ring as a subdirect product. A key part of the argument is the construction of a graph whose vertices are the subgroups of order $p$ and whose edges are determined by the covering.
V.I. Arnold (1971) constructed a simple normal form to which all complex matrices $B$ in a neighborhood $U$ of a given square matrix $A$ can be reduced by similarity transformations that smoothly depend on the entries of $B$. We calculate the radius of the neighborhood $U$. A.A. Mailybaev (1999, 2001) constructed a reducing similarity transformation in the form of Taylor series; we construct this transformation by another method. We extend Arnolds normal form to matrices over the field $mathbb Q_p$ of $p$-adic numbers and the field $mathbb F((T))$ of Laurent series over a field $mathbb F$.
These are significantly expanded lecture notes for the authors minicourse at MSRI in June 2012, as published in the MSRI lecture note series, with some minor additional corrections. In these notes, we give an example-motivated review of the deformation theory of associative algebras in terms of the Hochschild cochain complex as well as quantization of Poisson structures, and Kontsevichs formality theorem in the smooth setting. We then discuss quantization and deformation via Calabi-Yau algebras and potentials. Examples discussed include Weyl algebras, enveloping algebras of Lie algebras, symplectic reflection algebras, quasihomogeneous isolated hypersurface singularities (including du Val singularities), and Calabi-Yau algebras.
We study the higher Frobenius-Schur indicators of the representations of the Drinfeld double of a finite group G, in particular the question as to when all the indicators are integers. This turns out to be an interesting group-theoretic question. We show that many groups have this property, such as alternating and symmetric groups, PSL_2(q), M_{11}, M_{12} and regular nilpotent groups. However we show there is an irregular nilpotent group of order 5^6 with non-integer indicators.
To every involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation on a finite set $X$ there is a naturally associated finite solvable permutation group ${mathcal G}(X,r)$ acting on $X$. We prove that every primitive permutation group of this type is of prime order $p$. Moreover, $(X,r)$ is then a so called permutation solution determined by a cycle of length $p$. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic solutions.
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