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Composition-diamond lemma for modules

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Abstract

We investigate the relationship between the Gröbner-Shirshov bases in free associative algebras, free left modules and “double-free” left modules (that is, free modules over a free algebra). We first give Chibrikov’s Composition-Diamond lemma for modules and then we show that Kang-Lee’s Composition-Diamond lemma follows from it. We give the Gröbner-Shirshov bases for the following modules: the highest weight module over a Lie algebra sl 2, the Verma module over a Kac-Moody algebra, the Verma module over the Lie algebra of coefficients of a free conformal algebra, and a universal enveloping module for a Sabinin algebra. As applications, we also obtain linear bases for the above modules.

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P.R. China

    Yuqun Chen, Yongshan Chen & Chanyan Zhong

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  1. Yuqun Chen
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  2. Yongshan Chen
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  3. Chanyan Zhong
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Correspondence to Yuqun Chen.

Additional information

Supported by the NNSF of China (No. 10771077) and the NSF of Guangdong Province (No. 06025062).

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Chen, Y., Chen, Y. & Zhong, C. Composition-diamond lemma for modules. Czech Math J 60, 59–76 (2010). https://doi.org/10.1007/s10587-010-0018-2

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