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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.
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
Linking options:
https://www.mathnet.ru/eng/mzm10165https://doi.org/10.4213/mzm10165 https://www.mathnet.ru/eng/mzm/v93/i3/p407
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Abstract page: | 445 | Full-text PDF : | 142 | References: | 35 | First page: | 8 |
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