By this theorem, all dendriform dialgebra (resp. trialgebra) structures on V could be recovered from O-operators on the module (resp. on the algebra).
O-operators on associative algebras and dendriform algebras
Let (V, ≺, ≻, ·) be a dendriform trialgebra.
O-operators on associative algebras and dendriform algebras
Then it is straightforward to check that the dendriform trialgebra axioms of (V, ≺, ≻, · ) imply that (V, ·, L≻ , R≺ ) satis ﬁes all the axioms in Eq. (6) – (8) for a ( V, ∗)-bimodule k-algebra.
O-operators on associative algebras and dendriform algebras
Let (A, ≺, ≻, ·) be a dendriform trialgebra.
O-operators on associative algebras and dendriform algebras
Guo, Rota-Baxter algebras and dendriform algebras, J.
O-operators on associative algebras and dendriform algebras
Uchino, Quantum analogy of Poisson geometry, related dendriform algebras and Rota-Baxter operators, Lett.
O-operators on associative algebras and dendriform algebras
We generalize the well-known construction of dendriform dialgebras and trialgebras from Rota-Baxter algebras to a construction from O-operators.
O-operators on associative algebras and dendriform algebras
We then show that this construction from O-operators gives all dendriform dialgebras and trialgebras.
O-operators on associative algebras and dendriform algebras
Furthermore there are bijections between certain equivalence classes of invertible O-operators and certain equivalence classes of dendriform dialgebras and trialgebras.
O-operators on associative algebras and dendriform algebras
This paper shows that there is a close tie between two seemingly unrelated objects, namely Ooperators and dendriform dialgebras and trialgebras, generalizing and strengthening a previously established connection from Rota-Baxter algebras to dendriform algebras [1, 2, 13].
O-operators on associative algebras and dendriform algebras
On the other hand, with motivation from periodicity of algebraic K -theory and operads, dendriform dialgebras were introduced by Loday in the 1990s.
O-operators on associative algebras and dendriform algebras
P y = xP(y), x ≻P y = P( x)y, ∀ x, y ∈ R, deﬁne a dendriform dialgebra (R, ≺P , ≻P ).
O-operators on associative algebras and dendriform algebras
This deﬁnes a functor from the category of Rota-Baxter algeb ras of weight 0 to the category of dendriform dialgebras.
O-operators on associative algebras and dendriform algebras
These studies further suggested that there should be a close relationship between Rota-Baxter algebras and dendriform dialgebras.
O-operators on associative algebras and dendriform algebras
Then it is natural to ask whether every dendriform dialgebra and trialgebra could be derived from a Rota-Baxter algebra by a construction like Eq. (3).
O-operators on associative algebras and dendriform algebras
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