Universal enveloping algebra of a pair of compatible Lie brackets

TL;DR

This paper determines the structure and growth rate of the universal enveloping algebra for a pair of compatible Lie brackets using Gröbner-Shirshov bases, revealing it has polynomial growth of degree n+1.

Contribution

It provides a linear basis for the universal enveloping algebra of compatible Lie brackets and establishes its growth rate, extending classical Lie algebra results.

Findings

Linear basis of the universal enveloping algebra identified

Growth rate of the algebra is n+1 for n-dimensional cases

Application of Gröbner-Shirshov bases technique

Abstract

Applying the Poincare-Birkhoff-Witt property and the Groebner-Shirshov bases technique, we find the linear basis of the associative universal enveloping algebra in the sense of V. Ginzburg and M. Kapranov of a pair of compatible Lie brackets. We state that the growth rate of this universal enveloping over $n$-dimensional compatible Lie algebra equals $n+1$.