Some Examples of Nonassociative Coalgebras and Supercoalgebras

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.

Sub-Riemannian (2, 3, 5, 6)-Structures

We describe all Carnot algebras with growth vector (2, 3, 5, 6), their normal forms, an invariant that separates them, and a change of basis that transforms such an algebra into a normal form. For each normal form, Casimir functions and symplectic foliations on the Lie coalgebra are computed. An invariant and normal forms of left-invariant (2, 3, 5, 6)-distributions are described. A classification, up to isometries, of all left-invariant sub-Riemannian structures on (2, 3, 5, 6)-Carnot groups is obtained.

Wells sequences for abelian extensions of Lie coalgebras

For a Lie comodule [Formula: see text] of a Lie coalgebra [Formula: see text] and the Lie algebra [Formula: see text] (respectively, the group [Formula: see text]) of diagonal coderivations (respectively, Lie coalgebra automorphisms) of the semidirect sum [Formula: see text], we show that there is a representation of [Formula: see text] (respectively, [Formula: see text]) on the second cohomology group [Formula: see text], from which we construct obstruction classes of extensibility of pairs of coderivations (respectively Lie coalgebra automorphisms) and exact sequences of Wells type for abelian extensions of Lie coalgebras.