Jordan bialgebras of symmetric elements and Lie bialgebras

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

“…We let A be an associative algebra; a tensor r = ∑ a i ⊗ b i ∈ A ⊗ A is a solution of the AYBE [8,45,46] In 1983 [6], M.A. Semenov-Tyan-Shansky introduced the modified Yang-Baxter equation (MYBE).…”

“…In the 1980s, the deep connection between Lie RB-algebras and the classical Yang-Baxter equation (CYBE) was found [5,6]. In 2000, M. Aguiar showed [7] that a solution of the associative Yang-Baxter equation (AYBE) [8] gives rise to a structure of the associative RB-algebra. There are many applications of RB-operators in mathematical physics, combinatorics, number theory, and operads [9][10][11][12].…”

Pseudo-Euclidean Jordan Algebras

“…The Jordan Yang Baxter equation of J is C J (r) = 0 (see [16]). In this case, r is said an antisymmetric solution of The Jordan Yang Baxter equation or r is an antisymmetric r−matrix.…”

“…In this case, r is said an antisymmetric solution of The Jordan Yang Baxter equation or r is an antisymmetric r−matrix. In [16], it is proven that if r ∈ J ⊗ J is an antisymmetric r−matrix, then the pair (J, ∆ r ) is a Jordan bialgebra.…”

“…Recall that in [16] and [17], a Jordan algebra J is symplectic if it is endowed with an antisymmetric bilinear form not necessarily nondegenerate ω : J × J −→ K which satisfies (1). In the case where ω is nondegenerate, V.N Zhelyabin ([16], [17]) has obtained some results which similar to known results in the symplectic Lie algebras.…”