Jordan bialgebras and their relation to Lie bialgebras

Pre-jordan Algebras

“…The notion of Jordan D-bialgebra was introduced by Zhelyabin in [50] as an analogue of a Lie bialgebra (also see [51], [52], [53]). A class of Jordan D-bialgebras (coboundary cases) are obtained from the solutions of an algebraic equation in a Jordan algebra, which is an analogue of the classical Yang-Baxter equation (CYBE) in a Lie algebra ( [51], [53]).…”

“…According to [50] and [53], a Jordan D-bialgebra (J, ) is equivalent to the following structure: there is a Jordan algebra structure on the direct sum J ⊕ J * of the underlying vector spaces of J and J * such that both J and J * are subalgebras and the symmetric bilinear form on J ⊕ J * given by Eq. (1.11) is invariant.…”

Dual Coalgebras of Jordan Bialgebras and Superalgebras

“…A functional in A * is called a counity of a coalgebra (A, Δ) if a = a (a (1) )a (2) = a (a (2) )a (1) for an arbitrary element a in A; here Δ(a) = a a (1) ⊗ a (2) . Let (A, Δ) be a coalgebra.…”

“…The mapping Δ is called a comultiplication. Given an element a in a coalgebra (A, Δ) such that Δ(a) = i a 1i ⊗ a 2i , we will write Δ(a) = a a (1) ⊗ a (2) following Sweedler [8].…”

“…Lie bialgebras were introduced by V. G. Drinfel d in [1] as a formalism for describing solutions to the classical Yang-Baxter equation. In [2] the notion was defined of an associative and Jordan bialgebra in the sense of Drinfel d (D-bialgebra). In the same article, connection between some types of Jordan bialgebras and Lie bialgebras was studied.…”