2019/09/05 by Cantuba, Rafael Reno S.
#16R99 #16W70 #17B01 #17B35 #17B60 #17B65 #17B68 #17B70 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1909.02318
The universal enveloping algebra \mathscrU of a two-dimensional nonabelian Lie algebra L is a Lie algebra itself with the commutator as Lie bracket. There exists a presentation of \mathscrU with generators x,y and relation xy-yx=x such that the Lie subalgebra of \mathscrU generated by x,y is isomorphic to L, which is only a two-dimensional vector subspace of the infinite-dimensional \mathscrU. Much then of the Lie structure of \mathscrU is ubiquitous, yet unexamined when the characteristic of the scalar field is zero. In such a case, we show that there exists a linear complement of L in \mathscrU that contains an infinite-dimensional Lie subalgebra of \mathscrU for which we give a presentation by generators and relations. We extend this Lie subalgebra into a filtration of \mathscrU.