Integration and geometrization of Rota-Baxter Lie algebras

Rota-Baxter operators on cocommutative Hopf algebras

“…It is well known that if R : g → g is a Rota-Baxter operator of weight 1 on a Lie algebra g, then −R − id : g → g is again a Rota-Baxter operator of weight 1 on g. Similar results for groups was proved in [1]: if G is a group and B is a Rota-Baxter operator on G, then B : G → G defined by B(g) = g −1 B(g −1 ) is also a Rota-Baxter operator on G. For cocommutative Hopf algebras we can generalise these results:…”

“…Definition [1]. If R is a Rota-Baxter operator of weight 1 on a Lie algebra g, then the pair (g, {, }) is called the descendent Lie algebra of the Rota-Baxter Lie algebra (g, R).…”

“…Recently, in [1] it was introduced the notion of a Rota-Baxter operator (of weight 1) for groups. If G is a group, then a map B : G → G is called a Rota-Baxter operator on the group G if for all g, h ∈ G: B(g)B(h) = B(gB(g)hB(g) −1 ).…”

“…Another well-known result says that if a Lie algebra g splits into direct sum of two subalgebras g 1 and g 2 : g = g 1 ⊕ g 2 , then the map R defined as R(x 1 + x 2 ) = −x 2 , where x i ∈ g i , is a Rota-Baxter operator of weight 1. For groups similar result ( [1]) says that if a group G can be presented as a product of two subgroups G 1 and…”

Rota–Baxter Operators on Cocommutative Weak Hopf Algebras

“…Recently, the relationships between Rota-Baxter operators and Hopf algebras have attracted many researchers, such as Jian [9], Yu, Guo and Thibon [10], and Zheng, Guo, and Zhang [11]. In 2021, Guo, Lang, and Sheng gave the notion of a Rota-Baxter operator on a group [12], moreover, based on the above notion, Goncharov introduced the definition of a Rota-Baxter operator on a cocommutative Hopf algebra and proved that the Rota-Baxter operator on the universal enveloping algebra U(L) of a Lie algebra L is one to one corresponding to the Rota-Baxter operator on L [13]. As we know, a weak Hopf algebra is a generalization of a Hopf algebra.…”