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Register a new userThe Donald-Flanigan problem for finite reflection groups
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The Donald--Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra kH to a separable algebra. It is solved here for dihedral groups and for the classical Weyl groups (whose rational group algebras are also computed), leaving but six finite reflection groups with solutions unknown. We determine the structure of a wreath product of a group with a sum of central separable algebras and show that if there is a solution for H over k which is a sum of central separable algebras then there is also a solution for the wreath product of H with any symmetric group, abelian group, or dihedral group. The theorems suggested by the Donald-Flanigan conjecture and subsequently proven follow, we also show, from a geometric conjecture which although weaker for groups applies to a broader class of algebras than group algebras.
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We show that a Jordan-Holder theorem holds for appropriately defined composition series of finite dimensional Hopf algebras. This answers an open question of N. Andruskiewitsch. In the course of our proof we establish analogues of the Noether isomorphism theorems of group theory for arbitrary Hopf algebras under certain faithful (co)flatness assumptions. As an application, we prove an analogue of Zassenhaus butterfly lemma for finite dimensional Hopf algebras. We then use these results to show that a Jordan-Holder theorem holds as well for lower and upper composition series, even though the factors of such series may be not simple as Hopf algebras.
We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfeld twists of group algebras for the following groups: $A_n$, the alternating group on $n$ elements, with $n geq 5$; and $S_{2m}$, the symmetric group on $2m$ elements, with $m geq 4$ even. The twist for $A_n$ arises from a $2$-cocycle on the Klein four-group contained in $A_4$. The twist for $S_{2m}$ arises from a $2$-cocycle on a subgroup generated by certain transpositions which is isomorphic to $mathbb{Z}_2^m$. This provides more examples of complex semisimple Hopf algebras that can not be defined over number rings. As in the previous family known, these Hopf algebras are simple.
We continue the study of twisting of affine algebraic groups G (i.e., of Hopf 2-cocycles J for the function algebra O(G)), which was started in [EG1,EG2], and initiate the study of the associated one-sided twisted function algebras O(G)_J. We first show that J is supported on a closed subgroup H of G (defined up to conjugation), and that O(G)_J is finitely generated with center O(G/H). We then use it to study the structure of O(G)_J for connected nilpotent G. We show that in this case O(G)_J is a Noetherian domain, which is a simple algebra if and only if J is supported on G, and describe the simple algebras that arise in this way. We also use [EG2] to obtain a classification of Hopf 2-cocycles for connected nilpotent G, hence of fiber functors Rep(G)to Vect. Along the way we provide many examples, and at the end formulate several ring-theoretical questions about the structure of the algebras O(G)_J for arbitrary G.
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 the classical Weyl groups are infinite dimensional.
In this paper, we initiate the study of nondiagonal finite quasi-quantum groups over finite abelian groups. We mainly study the Nichols algebras in the twisted Yetter-Drinfeld module category $_{k G}^{k G}mathcal{YD}^Phi$ with $Phi$ a nonabelian $3$-cocycle on a finite abelian group $G.$ A complete clarification is obtained for the Nichols algebra $B(V)$ in case $V$ is a simple twisted Yetter-Drinfeld module of nondiagonal type. This is also applied to provide a complete classification of finite-dimensional coradically graded pointed coquasi-Hopf algebras over abelian groups of odd order and confirm partially the generation conjecture of pointed finite tensor categories due to Etingof, Gelaki, Nikshych and Ostrik.
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