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Almost split sequences of the quantum double of a finite group
Abstract
Let k be a field of characteristic p > 0, and G be a finite group of order divisible by p. We prove that the almost split sequences of the quantum double D(kG) can be constructed from those of group algebras, where the groups run over all centralizer subgroups of representatives of conjugate classes of G. As a special case, we give an application to the quantum double of dihedral groups.
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Let k be an algebraically closed field of odd characteristic p, and let
be the dihedral group of order 2n such that . Let
denote the quantum double of the group algebra . In this paper, we
describe the structures of all finite dimensional indecomposable left
-modules, equivalently, of all finite dimensional indecomposable
Yetter-Drinfeld -modules, and classify them.
Let k be an algebraically closed field of characteristic p > 0. We characterize the finite groups G for which the Drinfeld double D(kG) of the group algebra kG has the Chevalley property. We also show that this is the case if and only if the tensor product of every simple D(kG)-module with its dual is semisimple. The analogous result for the group algebra kG is also true, but its proof requires the classification of the finite simple groups. A further result concerns the largest Hopf ideal contained in the Jacobson radical of D(kG). We prove that this is generated by the augmentation ideal of kO p(Z(G)), where Z(G) is the center of G and O p(Z(G)) the largest p-subgroup of this center. © 2012 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.
This book is an introduction to the contemporary representation theory of Artin algebras, by three very distinguished practitioners in the field. Beyond assuming some first-year graduate algebra and basic homological algebra, the presentation is entirely self-contained, so the book is a suitable introduction for any mathematician (especially graduate students) to this field. The main aim of the book is to illustrate how the theory of almost split sequences is used in the representation theory of Artin algebras. However, other foundational aspects of the subject are developed. These results give concrete illustrations of some of the more abstract concepts and theorems. The book includes complete proofs of all theorems, and numerous exercises.
Let k be an algebraically closed field of characteristic p > 0. We characterize the finite groups G for which the Drinfeld double D(kG) of the group algebra kG has the Chevalley property. We also show that this is the case if and only if the tensor product of every simple D(kG)-module with its dual is semisimple. The analogous result for the group algebra kG is also true, but its proof requires the classification of the finite simple groups. A further result concerns the largest Hopf ideal contained in the Jacobson radical of D(kG). We prove that this is generated by the augmentation ideal of kOp(Z(G)), where Z(G) is the center of G and Op(Z(G)) the largest p-subgroup of this center.
Let H be a finite dimensional Hopf algebra and D(H) the associated ”quantum double” quasitriangular Hopf algebra of Drin-feld. Previously we showed that as an example of a double cross product of Hopf algebras HH acting on each other. We now show that if H is itself quasitriangular then D(H) is a semidirect"biproduct"in the sense of Radford: There is an algebra and coalgebra B such that as an ordinary smash product and smash co-product by H. The result arises naturally in the context of representation theoretic considerations for double cross products.
We present some results about the representation ring of the quantum double of a finite group over fields of arbitrary characteristic. We give a direct sum decomposition of this representation ring into ideals involving Green rings of subgroups. Given characters of such Green rings, we construct characters of the representation ring of the quantum double and show that all characters arise in this fashion. We prove that the characters thereby obtained from Brauer characters separate modules of the quantum double up to composition factors.
The Silences of the Archives, the Reknown of the Story.
The Martin Guerre affair has been told many times since Jean de Coras and Guillaume Lesueur published their stories in 1561. It is in many ways a perfect intrigue with uncanny resemblance, persuasive deception and a surprizing end when the two Martin stood face to face, memory to memory, before captivated judges and a guilty feeling Bertrande de Rols. The historian wanted to go beyond the known story in order to discover the world of the heroes. This research led to disappointments and surprizes as documents were discovered concerning the environment of Artigat’s inhabitants and bearing directly on the main characters thanks to notarial contracts. Along the way, study of the works of Coras and Lesueur took a new direction. Coming back to the affair a quarter century later did not result in finding new documents (some are perhaps still buried in Spanish archives), but by going back over her tracks, the historian could only be struck by the silences of the archives that refuse to reveal their secrets and, at the same time, by the possible openings they suggest, by the intuition that almost invisible threads link here and there characters and events.
Let k be a commutative ring, G′⊃G finite affine algebraic k-groups, and H′⊃H the dual Hopfalgebras of the affine algebras of G′ resp. G. The main results of this paper are: (I) If k is semilocal (e.g. k a field) there is an H′-linear, H∥H′-colinear, unitary, augmented isomorphism H→H∥H′{large circle} H′, where H∥H′ is the coalgebra belonging to G/G′. (II) If the k-submodule of the fixelements of (H∥H′)* is isomorphic to k (e.g. k principal or semilocal), then H′⊃H is a Frobeniusextension of the second kind.