Some examples of nonassociative coalgebras and supercoalgebras

TL;DR

This paper explores various classes of nonassociative coalgebras, providing examples, constructions, and studying their local finiteness and dual algebra properties, with implications for algebraic structures in characteristic zero.

Contribution

It introduces new examples and constructions of nonassociative coalgebras, including Novikov, Lie, and Jordan super-coalgebras, and analyzes their properties and duals.

Findings

Examples of non-locally finite differential coalgebras and super-coalgebras.

Construction of infinite-dimensional simple nonassociative coalgebras.

Dual algebras satisfy strong additional identities.

Abstract

Locally finiteness of some varieties of nonassociative coalgebras is studied and the Gelfand-Dorfman construction for Novikov coalgebras and the Kantor construction for Jordan super-coalgebras are given. We give examples of a non-locally finite differential coalgebra, Novikov coalgebra, Lie coalgebra, Jordan super-coalgebra, and right-alternative coalgebra. The dual algebra of each of these examples satisfies very strong additional identities. We also constructed examples of an infinite dimensional simple differential coalgebra, Novikov coalgebra, Lie coalgebra, and Jordan super-coalgebra over a field of characteristic zero.