Universal Enveloping Commutative Rota–Baxter Algebras of Pre- and Post-Commutative Algebras

Abstract

Universal enveloping commutative Rota-Baxter algebras of pre-and post-commutative algebras are constructed. The pair of varieties (RB λ Com, postCom) is proved to be a Poincaré-Birkhoff-Witt-pair (PBW)-pair and the pair (RBCom, preCom) is proven not to be.

Poincare-Birkhoff-Witt theorem for pre-Lie and postLie algebras

“…In 2013 [25], given a variety Var, it was proved that every pre-Var-algebra (post-Varalgebra) injectively embeds into its universal enveloping Var-RB-algebra of weight λ = 0 (λ = 0). Further, author constructed universal enveloping RB-algebra for a given pre-or postalgebra in commutative [20], associative [21], and Lie [22] cases. In the associative case it gave an answer to the question of L. Guo [27, p. 148].…”

Endofunctors and Poincaré-Birkhoff-Witt theorems

“…We argue that our result, being a necessary and sufficient statement, should be regarded as the bare-bones framework for studying the PBW property; as such, it provides one with a unified approach to numerous PBW type results proved -sometimes by very technical methods-in the literature, see e.g. [16,17,31,35,36,48,49,53,54,57]. Most of those PBW type theorems tend to utilise something extrinsic; e.g., in the case of Lie algebras, one may consider only Lie algebras associated to Lie groups and identify the universal enveloping algebra with the algebra of distributions on the group supported at the unit element (see [55], this is probably the closest in spirit to the original proof of Poincaré [50]), or use the additional coalgebra structure on the universal enveloping algebra (like in the proof of Cartier [12], generalised by Loday in [41] who defined a general notion of a "good triple of operads").…”

On Transcendental Numbers: New Results and a Little History

“…In some special issues on Hopf algebras, quantum groups and Yang-Baxter equations, several papers [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], as well the feature paper [23], covered many topics related to the Yang-Baxter equation, ranging from mathematical physics to Hopf algebras, from Azumaya Monads to quantum computing, and from Mathematical Logic to Rota-Baxter equations.…”