Gröbner–Shirshov bases for symmetric brace algebras

Abstract

In this paper, we develop Gröbner–Shirshov basis theory for symmetric brace algebras. As applications, firstly, we give an alternative proof that the category of symmetric brace algebras is isomorphic to the category of pre-Lie algebras, secondly, we prove that each pre-Lie algebra can be embedded into a simple pre-Lie algebra, and finally, we present a new linear basis for the universal pre-Lie algebra of a Lie algebra.

Keywords:

• Gröbner–Shirshov basis
• pre-Lie algebra
• symmetric brace algebra

AMS MATHEMATICS SUBJECT CLASSIFICATION (2020):

• 17A61
• 16S15
• 17A30
• 17D25

Acknowledgment

The authors would like to thank the anonymous referee for helpful comments.

Additional information

Funding

This work was financially supported by the NNSF of China [No. 11501237], the NSF of Guangdong Province [No. 2016A030310099], the Characteristic Innovation Project of Guangdong Provincial Department of Education [2020KTSCX145], and the Huizhou University [hzuxl201523, hzu201704, hzu201804].