Universal enveloping algebra of a pair of compatible Lie brackets

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 $n+1$.

9 p.; v2: the defining relations of the universal enveloping algebra in the sense of V. Ginzburg and M. Kapranov of a pair of compatible Lie brackets are corrected; v3: Remarks 2 and 3 are added

References in corpus (3)

• Gröbner-Shirshov bases and their calculation
• Character formulas for the operad of two compatible brackets and for the bihamiltonian operad
• Operads of compatible structures and weighted partitions