Dual Coalgebras of Jordan Bialgebras and Superalgebras

Abstract

W. Michaelis showed for Lie bialgebras that the dual coalgebra of a Lie algebra is a Lie bialgebra. In the present article we study an analogous question in the case of Jordan bialgebras. We prove that a simple infinite-dimensional Jordan superalgebra of vector type possesses a nonzero dual coalgebra. Thereby, we demonstrate that the hypothesis formulated by W. Michaelis for Lie coalgebras fails in the case of Jordan supercoalgebras.

Some Examples of Nonassociative Coalgebras and Supercoalgebras

“…for all n ≥ 0. Notice that an example of a Jordan super-coalgebra without finite dimensional subcoagebras was constructed in [23] 6. Non locally finite right alternative coalgebra…”

Some examples of nonassociative coalgebras and supercoalgebras

“…Proof. Let B be a nonzero subcoalgebra of (J(C, d), ∆ J ) and let Notice that an example of a Jordan super-coalgebra without finite dimensional subcoagebras was constructed in [23] 6. Non locally finite right alternative coalgebra…”