The Representation Ring of the Twisted Quantum Double of a Finite Group

1996 ◽  
Vol 48 (6) ◽  
pp. 1324-1338 ◽  
Author(s):  
S. J. Witherspoon
Keyword(s):  
Finite Group ◽  
Group Algebras ◽  
Quantum Double ◽  
Grothendieck Ring ◽  
Representation Ring ◽  
Twisted Group ◽  
A Finite Group

AbstractWe provide an isomorphism between the Grothendieck ring of modules of the twisted quantum double of a finite group, and a product of centres of twisted group algebras of centralizer subgroups. It follows that this Grothendieck ring is semisimple. Another consequence is a formula for the characters of this ring in terms of representations of twisted group algebras of centralizer subgroups.

scholarly journals Twisted group algebras and their representations

1964 ◽  
Vol 4 (2) ◽  
pp. 152-173 ◽  
Author(s):  
S. B. Conlon
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Group Algebras ◽  
Linear Combinations ◽  
Twisted Group Algebra ◽  
Formula One ◽  
Twisted Group ◽  
Image Position ◽  
A Finite Group ◽  
Group Product

Let be a finite group, a field. A twisted group algebra A() on over is an associative algebra whose elements are the formal linear combinations and in which the product (A)(B) is a non-zero multiple of (AB), where AB is the group product of A, B ∈: . One gets the ordinary group algebra () by taking each fA, B ≠ 1.

scholarly journals Quantum double finite group algebras and their representations

1993 ◽  
Vol 48 (2) ◽  
pp. 275-301 ◽  
Author(s):  
M.D. Gould
Keyword(s):  
Finite Group ◽  
Explicit Construction ◽  
Unitary Representations ◽  
Group Algebras ◽  
Baxter Equation ◽  
Irreducible Matrix ◽  
Matrix Representations ◽  
Quantum Double ◽  
Induced Representations ◽  
A Finite Group

The quantum double construction is applied to the group algebra of a finite group. Such algebras are shown to be semi-simple and a complete theory of characters is developed. The irreducible matrix representations are classified and applied to the explicit construction of R-matrices: this affords solutions to the Yang-Baxter equation associated with certain induced representations of a finite group. These results are applied in the second paper of the series to construct unitary representations of the Braid group and corresponding link polynomials.

The Representation Ring and the Centre of a Hopf Algebra

1999 ◽  
Vol 51 (4) ◽  
pp. 881-896 ◽  
Author(s):  
Sarah J. Witherspoon
Keyword(s):  
Finite Group ◽  
Hopf Algebra ◽  
Character Table ◽  
Quantum Double ◽  
Representation Ring ◽  
Finite Dimensional ◽  
Dual Character ◽  
Structure Constants ◽  
A Finite Group ◽  
Cocommutative Hopf Algebra

AbstractWhen H is a finite dimensional, semisimple, almost cocommutative Hopf algebra, we examine a table of characters which extends the notion of the character table for a finite group. We obtain a formula for the structure constants of the representation ring in terms of values in the character table, and give the example of the quantum double of a finite group. We give a basis of the centre of H which generalizes the conjugacy class sums of a finite group, and express the class equation of H in terms of this basis. We show that the representation ring and the centre of H are dual character algebras (or signed hypergroups).

scholarly journals ON THE MORITA REDUCED VERSIONS OF SKEW GROUP ALGEBRAS OF PATH ALGEBRAS

2020 ◽  
Vol 71 (3) ◽  
pp. 1009-1047
Author(s):  
Patrick Le Meur
Keyword(s):  
Finite Group ◽  
Monoidal Category ◽  
Path Algebra ◽  
Group Algebras ◽  
Explicit Formulas ◽  
Morita Equivalent ◽  
Path Algebras ◽  
A Finite Group ◽  
Skew Group Algebras ◽  
Skew Group Algebra

Abstract Let $R$ be the skew group algebra of a finite group acting on the path algebra of a quiver. This article develops both theoretical and practical methods to do computations in the Morita-reduced algebra associated to $R$. Reiten and Riedtmann proved that there exists an idempotent $e$ of $R$ such that the algebra $eRe$ is both Morita equivalent to $R$ and isomorphic to the path algebra of some quiver, which was described by Demonet. This article gives explicit formulas for the decomposition of any element of $eRe$ as a linear combination of paths in the quiver described by Demonet. This is done by expressing appropriate compositions and pairings in a suitable monoidal category, which takes into account the representation theory of the finite group.

scholarly journals The Grothendieck ring of a finite group

Topology ◽  
1963 ◽  
Vol 2 (1-2) ◽  
pp. 85-110 ◽  
Author(s):  
Richard G Swan
Keyword(s):  
Finite Group ◽  
Grothendieck Ring ◽  
A Finite Group

The nilpotency index of the radicals of group algebras of finite groups whose Sylow 3-subgroups are extra-special of order 27 of exponent 3

1988 ◽  
Vol 108 (1-2) ◽  
pp. 117-132
Author(s):  
Shigeo Koshitani
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Simple Group ◽  
Finite Groups ◽  
Jacobson Radical ◽  
Group Algebras ◽  
Nilpotency Index ◽  
Characteristic 3 ◽  
A Finite Group

SynopsisLet J(FG) be the Jacobson radical of the group algebra FG of a finite groupG with a Sylow 3-subgroup which is extra-special of order 27 of exponent 3 over a field F of characteristic 3, and let t(G) be the least positive integer t with J(FG)t = 0. In this paper, we prove that t(G) = 9 if G has a normal subgroup H such that (|G:H|, 3) = 1 and if H is either 3-solvable, SL(3,3) or the Tits simple group 2F4(2)'.

scholarly journals A classification of exceptional components in group algebras over abelian number fields

2016 ◽  
Vol 15 (05) ◽  
pp. 1650092
Author(s):  
Andreas Bächle ◽  
Mauricio Caicedo ◽  
Inneke Van Gelder
Keyword(s):  
Finite Group ◽  
Unit Group ◽  
Number Fields ◽  
Group Algebras ◽  
Matrix Rings ◽  
Division Rings ◽  
Wedderburn Decomposition ◽  
A Finite Group

When considering the unit group of [Formula: see text] ([Formula: see text] the ring of integers of an abelian number field [Formula: see text] and a finite group [Formula: see text]) certain components in the Wedderburn decomposition of [Formula: see text] cause problems for known generic constructions of units; these components are called exceptional. Exceptional components are divided into two types: type 1 is division rings, type 2 is [Formula: see text]-matrix rings. For exceptional components of type 1 we provide infinite classes of division rings by describing the seven cases of minimal groups (with respect to quotients) having those division rings in their Wedderburn decomposition over [Formula: see text]. We also classify the exceptional components of type 2 appearing in group algebras of a finite group over number fields [Formula: see text] by describing all 58 finite groups [Formula: see text] having a faithful exceptional Wedderburn component of this type in [Formula: see text].

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