PBW-Pairs of Varieties of Linear Algebras

Михалев, А. В.; Shestakov, I. P.

doi:10.1080/00927872.2012.720867

Cited by 42 publications

(49 citation statements)

References 28 publications

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“…First of all, we note that all the rewriting rules are consistent with the defining relations of RBLie 1 . Indeed, the rewriting rules (16) and (17) are simply the definitions of the two shuffle post-Lie operations, each of the rewriting rules (18) and (19) is in fact equivalent to the Rota-Baxter relation (2), and the remaining relations, as we already mentioned, are defining relations of the post-Lie operad which are known to hold in RBLie 1 . Second, this rewriting system is terminating.…”

Section: Pbw Theorem For Universal Enveloping Rota-baxter Lie Algebramentioning

confidence: 98%

Functorial PBW theorems for post-Lie algebras

Dotsenko

2020

Communications in Algebra

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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.

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Section: Pbw Theorem For Universal Enveloping Rota-baxter Lie Algebramentioning

confidence: 98%

Functorial PBW theorems for post-Lie algebras

Dotsenko

2020

Communications in Algebra

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“…Brace algebras have strong connections with other important classes of algebras. For instance, brace algebras are used to prove Milnor-Moore type theorems for some Hopf algebras [28,29]; a free brace algebra has also a free pre-Lie algebra structure (and thus has a free Lie structure) [11]; the pair of varieties (Brace, Pre-Lie) is a PBW-pair (in the sense of [27]) [19].…”

Section: Introductionmentioning

confidence: 99%

The Freiheitssatz and automorphisms for free brace algebras

Liu

Zhao

2019

Communications in Algebra

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“…Applying induction on r, we give the more detailed RHS of Equation (29) We give the second row of Equation (31) Subtracting Equation (28) from the sum of Equations (30), (32) and (34), we obtain zero by induction.…”

Section: Lemmamentioning

confidence: 99%

“…In [32], the notion of a PBW-pair that generalizes the relation between associative and Lie algebras given by the famous Poincaré-Birkhoff-Witt theorem was introduced.…”

Section: Pbw-pair Of Varietiesmentioning

confidence: 99%