Affine actions on Lie groups and post-Lie algebra structures

Burde, Dietrich; Dekimpe, Karel; Vercammen, Kim

doi:10.1016/j.laa.2012.04.007

Cited by 54 publications

(107 citation statements)

References 34 publications

Supporting

Mentioning

105

Contrasting

Order By: Relevance

“…First of all, we note that all the rewriting rules are consistent with the defining relations of RBAss 1 . Indeed, the rewriting rules (5) and (6) are simply the definitions of the two tridendriform operations, each of the rewriting rules (7) and (8) is in fact equivalent to the Rota-Baxter relation (4), and the remaining relations, as we already mentioned, are defining relations of the tridendriform operad which are known to hold in RBAss 1 . Second, this rewriting system is terminating.…”

Section: Algebrasmentioning

confidence: 98%

Functorial PBW theorems for post-Lie algebras

Dotsenko

2020

Communications in Algebra

View full text Add to dashboard Cite

Using the categorical approach to Poincaré-Birkhoff-Witt type theorems from our previous work with Tamaroff, we prove three such theorems: for universal enveloping Rota-Baxter algebras of tridendriform algebras, for universal enveloping Rota-Baxter Lie algebras of post-Lie algebras, and for universal enveloping tridendriform algebras of post-Lie algebras. Similar results, though without functoriality of the PBW isomorphisms, were recently obtained by Gubarev. Our methods are completely different and mainly rely on methods of rewriting theory for shuffle operads.2010 Mathematics Subject Classification. 17B35 (Primary), 16B50, 18D50, 68Q42 (Secondary). 1 It is perhaps worth mentioning that we do not consider in this paper another important kind of universal enveloping algebras of post-Lie algebras, the so called D-algebras prominent in the Lie-Butcher calculus [11,20]. Those algebras are obtained via an adjunction not arising from change of algebraic structure and, as a consequence, are a bit different; a PBW type theorem for them is true on the nose because of the Lie-theoretic nature of the definition.Acknowledgements. I am grateful to Vsevolod Gubarev for a discussion of results of [14], and to Murray Bremner for his comments on the paper [9], and particularly for a query as to how results of that paper may be applied to post-Lie algebras.

show abstract

Section: Algebrasmentioning

confidence: 98%

Functorial PBW theorems for post-Lie algebras

Dotsenko

2020

Communications in Algebra

View full text Add to dashboard Cite

show abstract

“…Let K denote a field of characteristic zero. We have defined a post-Lie algebra structure on a pair of Lie algebras (g, n) over K in [10] as follows: [ , ]) and n = (V, { , }) be two Lie brackets on a vector space V over K. A post-Lie algebra structure, or PA-structure on the pair (g, n) is a K-bilinear product x · y satisfying the identities…”

Section: Preliminariesmentioning

confidence: 99%

“…x · {y, z} = {x · y, z} + {y, x · z}, i.e., −x · y is an LR-structure on the Lie algebra n. For details see [10].…”

Section: Preliminariesmentioning

confidence: 99%

“…Post-Lie algebras and post-Lie algebra structures arise in many different contexts. We have introduced these structures in [10] to characterize simply transitive nil-affine actions of a nilpotent Lie group G on another nilpotent Lie group N. This plays an important role in the theory of nil-affine crystallographic groups and complete nil-affinely flat manifolds. Post-Lie algebras arise there as a natural common generalization of pre-Lie algebras [20,21,24,4,5,6] and LRalgebras [8,9], and the geometric questions can be formulated on the level of post-Lie algebras.…”

Section: Introductionmentioning

confidence: 99%

“…This is in general a very hard question and the answer depends very much on the algebraic properties of the two given Lie algebras. For a survey on the results and open problems see [7,10,11]. An important class of post-Lie algebra structures is given by commutative structures, so-called CPA-structures.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Commutative post-Lie algebra structures on nilpotent Lie algebras and Poisson algebras

Burde

Ender

2020

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

show abstract

Post-Lie Algebras, Factorization Theorems and Isospectral Flows

Ebrahimi‐Fard

Mencattini

2018

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

In these notes we review and further explore the Lie enveloping algebra of a post-Lie algebra. From a Hopf algebra point of view, one of the central results, which will be recalled in detail, is the existence of a second Hopf algebra structure. By comparing group-like elements in suitable completions of these two Hopf algebras, we derive a particular map which we dub post-Lie Magnus expansion. These results are then considered in the case of Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined in terms of solutions of modified classical Yang-Baxter equation. In this context, we prove a factorization theorem for group-like elements. An explicit exponential solution of the corresponding Lie bracket flow is presented, which is based on the aforementioned post-Lie Magnus expansion.

show abstract

Affine actions on Lie groups and post-Lie algebra structures

Abstract: This survey on crystallographic groups, geometric structures on Lie groups and associated algebraic structures is based on a lecture given in the Ostrava research seminar in 2017.

Cited by 54 publications

References 34 publications

Functorial PBW theorems for post-Lie algebras

Functorial PBW theorems for post-Lie algebras

Commutative post-Lie algebra structures on nilpotent Lie algebras and Poisson algebras

Post-Lie Algebras, Factorization Theorems and Isospectral Flows

Contact Info

Product

Resources

About