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Keywords:
Gröbner-Shirshov basis; module; Lie algebra; Kac-Moody algebra; conformal algebra; Sabinin algebra
Gröbner-Shirshov basis; module; Lie algebra; Kac-Moody algebra; conformal algebra; Sabinin algebra
Summary:
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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