Finite-dimensional Lie subalgebras of the universal enveloping algebra of sl(2,ℂ)
Abstract
In this work we provide the complete classification of all non-zero finite-dimensional complex Lie subalgebras of the universal enveloping algebra of the special linear Lie algebra sl(2,ℂ). That is, we show that every such nilpotent Lie subalgebra is abelian and every non-solvable Lie subalgebra is reductive, with semisimple component isomorphic to sl(2,ℂ) itself. Every non-nilpotent solvable Lie subalgebra is the direct sum of an abelian center and a member of the family of solvable Lie algebras r(d,m), parameterized by a vector d whose entries are strictly increasing positive integers that have greatest common divisor one together with a positive integer vector m of the same length recording their multiplicities. Moreover, we discuss the corresponding classification in a real skew-hermitian setting and its relation to the realizations in the single- and two-mode Weyl algebras.
Related
- Finite free position of maximal abelian ∗-subalgebras of the matrix algebra
- Maximal subalgebras in the three exceptional simple finite-dimensional Lie superalgebras
- Ideals of homomorphic images of the enveloping algebra of the Witt algebra
- Universal Enveloping Algebras of Hom-Lie Algebras with α-Derivations
- The Serre--Hochschild plane of a finite-dimensional algebra
BibTeX
@misc{heib2026finitedimensional,
title = {{Finite-dimensional Lie subalgebras of the universal enveloping algebra of $\mathfraksl(2,\mathbbC)$}},
author = {Tim Heib},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04001},
eprint = {2610.04001},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04001},
}
RIS
TY - UNPB
TI - Finite-dimensional Lie subalgebras of the universal enveloping algebra of sl(2,ℂ)
AU - Tim Heib
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.04001
UR - https://arxiv.org/abs/2610.04001
AB - In this work we provide the complete classification of all non-zero finite-dimensional complex Lie subalgebras of the universal enveloping algebra of the special linear Lie algebra sl(2,ℂ). That is, we show that every such nilpotent Lie subalgebra is abelian and every non-solvable Lie subalgebra is reductive, with semisimple component isomorphic to sl(2,ℂ) itself. Every non-nilpotent solvable Lie subalgebra is the direct sum of an abelian center and a member of the family of solvable Lie algebras r(d,m), parameterized by a vector d whose entries are strictly increasing positive integers that have greatest common divisor one together with a positive integer vector m of the same length recording their multiplicities. Moreover, we discuss the corresponding classification in a real skew-hermitian setting and its relation to the realizations in the single- and two-mode Weyl algebras.
ER -