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Mat. Zametki, 2013, Volume 93, Issue 3, Pages 407–412 (Mi mz10165)  

The Complexity of Crossed Products

Ling Liua, Bing-Liang Shenb

a Zhejiang Normal University
b Shanghai University of Finance & Economics, Zhejiang College

Abstract: Let $H$ be a finite-dimensional Hopf algebra, $A$ be a finite-dimensional algebra measured by $H$ and $A\mathbin{#_\sigma}H$ be a crossed product. In this paper, we first show that if $H$ is semisimple as well as its dual $H^*$, then the complexity of $A\mathbin{#_\sigma} H$ is equal to that of $A$. Furthermore, we prove that the complexity of a finite-dimensional Hopf algebra $H$ is equal to the complexity of the trivial module $_Hk$. As an application, we prove that the complexity of Sweedler's 4-dimensional Hopf algebra $H_4$ is equal to $1$.

Keywords: crossed product, complexity, trivial module, Sweedler's 4-dimensional Hopf algebra.

DOI: https://doi.org/10.4213/mzm10165

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English version:
Mathematical Notes, 2013, 93:3, 426–430

Bibliographic databases:

UDC: 512.667
Received: 15.06.2011

Citation: Ling Liu, Bing-Liang Shen, “The Complexity of Crossed Products”, Mat. Zametki, 93:3 (2013), 407–412; Math. Notes, 93:3 (2013), 426–430

Citation in format AMSBIB
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