Abstract
Universal enveloping commutative Rota-Baxter algebras of pre- and postcommutative algebras are constructed. We prove that the pair of varieties (commutative Rota-Baxter algebras of nonzero weight,postcommutative algebras) is a PBW-pair and the pair (commutative Rota-Baxter algebras of zero weight,precommutative algebras) is not.
Highlights
Any linear operator R defined on an algebra A over the field F is called a Rota–Baxter operator (RB-operator) of weight λ ∈ F if it satisfies the relation
For r = ∑ ai ⊗ bi ∈ g ⊗ g, we introduce the classical Yang—Baxter equation (CYBE) given by A.A
In [5,6], it was shown that given a skew-symmetric solution of the classical Yang–Baxter equation (CYBE) on g, a linear map R : g → g defined as
Summary
Any linear operator R defined on an algebra A over the field F is called a Rota–Baxter operator (RB-operator) of weight λ ∈ F if it satisfies the relation. In [27,28,29], post-associative, -commutative, and -Lie algebras were introduced All of these have an additional product and satisfy certain identities. On the basis of the last result, we have a problem: to construct the universal enveloping RB-algebra of a variety Var for pre- and post-Var-algebras Another related problem is the following: whether the pairs of varieties (RBVar, preVar) and (RBλ Var, postVar), λ 6= 0, are Poincaré–Birkhoff–Witt (PBW)-pairs [32]. Vallette in [29]; for example, see [43] This is why we prefer to name dendriform algebras as pre-associative algebras and Zinbiel algebras as pre-commutative algebras (analogously for the varieties of post-algebras)
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