Universal enveloping Lie Rota-Baxter algebras of pre-Lie and post-Lie algebras. (English. Russian original) Zbl 1472.17047
Algebra Logic 58, No. 1, 1-14 (2019); translation from Algebra Logika 58, No. 1, 3-21 (2019).
Let \(L\) be a vector space over a field \(F\). Define a bilinear product \(xy\) in \(L\) such that for every \(x_1\), \(x_2\), \(x_3\in L\) one has \((x_1x_2)x_3 - x_1(x_2x_3) = (x_2x_1)x_3 - x_2(x_1x_3)\), then \(L\) is a pre-Lie algebra. The name comes from the fact that substituting the product in \(L\) by \([x,y]=xy-yx\) one gets a Lie algebra.
A post-Lie algebra is a vector space \(L\) together with two bilinear operations \([x,y]\) and \(xy\) such that \(L\) is a Lie algebra considered with \([x,y]\), and \((x_1x_2)x_3 - x_1(x_2x_3) -(x_2x_1)x_3 + x_2(x_1x_3) = [x_2,x_1]x_3\), and moreover \(x_1[x_2,x_3] = [x_1x_2,x_3]+ [x_2,x_1x_3]\), for every \(x_1\), \(x_2\), \(x_3\in L\).
The author studies Rota-Baxter algebras (that is algebras equipped with a Rota-Baxter operator). It is proved in the paper that the pre-Lie and the post-Lie algebras have universal enveloping Lie Rota-Baxter algebras, moreover the authors gives a direct construction of these universal enveloping algebras. Furthermore he proves that the variety of all Lie Rota-Baxter algebras is not a Schreier variety. Additionally he proves that the variety of Lie Rota-Baxter algebras and the one of the pre-Lie algebras are related as in the Poincaré-Birkhoff-Witt theorem (that is they form a PBW-pair). A similar result is proved for the varieties of all \(\lambda\)-Lie Rota-Baxter algebras and all post-Lie algebras.
A post-Lie algebra is a vector space \(L\) together with two bilinear operations \([x,y]\) and \(xy\) such that \(L\) is a Lie algebra considered with \([x,y]\), and \((x_1x_2)x_3 - x_1(x_2x_3) -(x_2x_1)x_3 + x_2(x_1x_3) = [x_2,x_1]x_3\), and moreover \(x_1[x_2,x_3] = [x_1x_2,x_3]+ [x_2,x_1x_3]\), for every \(x_1\), \(x_2\), \(x_3\in L\).
The author studies Rota-Baxter algebras (that is algebras equipped with a Rota-Baxter operator). It is proved in the paper that the pre-Lie and the post-Lie algebras have universal enveloping Lie Rota-Baxter algebras, moreover the authors gives a direct construction of these universal enveloping algebras. Furthermore he proves that the variety of all Lie Rota-Baxter algebras is not a Schreier variety. Additionally he proves that the variety of Lie Rota-Baxter algebras and the one of the pre-Lie algebras are related as in the Poincaré-Birkhoff-Witt theorem (that is they form a PBW-pair). A similar result is proved for the varieties of all \(\lambda\)-Lie Rota-Baxter algebras and all post-Lie algebras.
Reviewer: Plamen Koshlukov (Campinas)
MSC:
| 17B35 | Universal enveloping (super)algebras |
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 17B01 | Identities, free Lie (super)algebras |
Keywords:
pre-Lie algebra; post-Lie algebra; Rota-Baxter algebra; universal enveloping algebra; Lyndon-Shirshov word; PBW-pair of varieties; Schreier variety; partially commutative Lie algebraReferences:
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