On twisted group algebras of OTP representation type over the ring of $p$-adic integers

Leonid F. Barannyk¹ ; Dariusz Klein¹

¹ Institute of Mathematics Pomeranian University of Słupsk Arciszewskiego 22d 76-200 Słupsk, Poland

Colloquium Mathematicum, Tome 143 (2016) no. 2, pp. 209-235

Résumé

Let $\hat{\mathbb{Z}}_p$ be the ring of $p$-adic integers, $U(\hat{\mathbb{Z}}_p)$ the unit group of $\hat{\mathbb{Z}}_p$ and $G=G_p\times B$ a finite group, where $G_p$ is a $p$-group and $B$ is a $p’$-group. Denote by $\hat{\mathbb{Z}}_p^\lambda G$ the twisted group algebra of $G$ over $\hat{\mathbb{Z}}_p$ with a $2$-cocycle $\lambda\in Z^2(G,U(\hat{\mathbb{Z}}_p))$. We give necessary and sufficient conditions for $\hat{\mathbb{Z}}_p^\lambda G$ to be of OTP representation type, in the sense that every indecomposable $\hat{\mathbb{Z}}_p^\lambda G$-module is isomorphic to the outer tensor product $V\mathbin{\#} W$ of an indecomposable $\hat{\mathbb{Z}}_p^\lambda G_p$-module $V$ and an irreducible $\hat{\mathbb{Z}}_p^\lambda B$-module $W$.

Détail

DOI : 10.4064/cm6700-1-2016

Keywords: hat mathbb ring p adic integers hat mathbb unit group hat mathbb times finite group where p group group denote hat mathbb lambda twisted group algebra hat mathbb cocycle lambda hat mathbb necessary sufficient conditions hat mathbb lambda otp representation type sense every indecomposable hat mathbb lambda g module isomorphic outer tensor product mathbin indecomposable hat mathbb lambda p module irreducible hat mathbb lambda b module