On the Lie Enveloping Algebra of a Post-Lie Algebra
By Kurusch Ebrahimi-Fard, Alexander Lundervold, H. Munthe-Kaas. Journal of Lie theory, 2014
74 citations recorded by the source corpus.
We consider pairs of Lie algebras $g$ and $\bar{g}$, defined over a common vector space, where the Lie brackets of $g$ and $\bar{g}$ are related via a post-Lie algebra structure. The latter can be extended to the Lie enveloping algebra $U(g)$. This permits us to define another associative product on $U(g)$, which gives rise to a Hopf algebra isomorphism between $U(\bar{g})$ and a new Hopf algebra assembled from $U(g)$ with the new product. For the free post-Lie algebra these constructions provide a refined understanding of a fundamental Hopf algebra appearing in the theory of numerical integration methods for differential equations on manifolds. In the pre-Lie setting, the algebraic point of view developed here also provides a concise way to develop Butcher's order theory for Runge--Kutta methods.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/80835367c884dd679c0a0b58d1fe571c42ca129c
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Citations in the corpus
- Rota-Baxter operators on cocommutative Hopf algebras. (2020, unknown): “In section 4 we first show, that this extension of a post-Lie algebra structure to the universal enveloping algebra U(g) (that was found in [13]) can be defined using the Rota-Baxter Hopf operator B : U(g) 7→ U(g) that is the extension of R.”
- Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation (2023, background): “See [22, 36, 39, 45] for more studies on the universal enveloping algebras of post-Lie algebras.”