Gr\"obner-Shirshov bases theory and extensions of Leibniz superalgebras
Yuxiu Bai, Yuqun Chen
TL;DR
This paper develops Gr"obner-Shirshov bases for Leibniz superalgebras, establishing uniqueness of reduced bases, constructing linear bases for free metabelian cases, and characterizing algebra extensions.
Contribution
It introduces a method to find unique reduced Gr"obner-Shirshov bases for Leibniz superalgebras and applies it to construct bases and characterize extensions.
Findings
Unique reduced Gr"obner-Shirshov bases exist for all graded ideals.
Constructed linear bases for free metabelian Leibniz superalgebras.
Provided a complete characterization of Leibniz algebra extensions.
Abstract
In this paper, we elaborate Gr\"obner-Shirshov bases method for Leibniz (super)algebras. We show that there is a unique reduced Gr\"obner-Shirshov basis for every (graded) ideal of a free Leibniz (super)algebra. As applications, we construct linear bases of free metabelian Leibniz superalgebras and new linear bases of free metabelian Lie algebras. We present a complete characterization of extensions of a Leibniz (super)algebra by another Leibniz (super)algebra, where the former is presented by generators and relations.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology