Composition-diamond lemma for modules

Yuqun Chen; Yongshan Chen; Chanyan Zhong

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 1, page 59-76
  • ISSN: 0011-4642

Abstract

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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 s l 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.

How to cite

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Chen, Yuqun, Chen, Yongshan, and Zhong, Chanyan. "Composition-diamond lemma for modules." Czechoslovak Mathematical Journal 60.1 (2010): 59-76. <http://eudml.org/doc/37988>.

@article{Chen2010,
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.},
author = {Chen, Yuqun, Chen, Yongshan, Zhong, Chanyan},
journal = {Czechoslovak Mathematical Journal},
keywords = {Gröbner-Shirshov basis; module; Lie algebra; Kac-Moody algebra; conformal algebra; Sabinin algebra; Gröbner-Shirshov bases; free associative algebras; Lie algebras; Kac-Moody algebras; conformal algebras; Sabinin algebras; free left modules},
language = {eng},
number = {1},
pages = {59-76},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Composition-diamond lemma for modules},
url = {http://eudml.org/doc/37988},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Chen, Yuqun
AU - Chen, Yongshan
AU - Zhong, Chanyan
TI - Composition-diamond lemma for modules
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 1
SP - 59
EP - 76
AB - 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.
LA - eng
KW - Gröbner-Shirshov basis; module; Lie algebra; Kac-Moody algebra; conformal algebra; Sabinin algebra; Gröbner-Shirshov bases; free associative algebras; Lie algebras; Kac-Moody algebras; conformal algebras; Sabinin algebras; free left modules
UR - http://eudml.org/doc/37988
ER -

References

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