A nonisomorphism theorem for cofree Lie coalgebras

Abstract

In the first paragraph of this section we will give a brief overview of this work. The remainder of this section contains formal definitions and details omitted from the first paragraph. Throughout this work, all vector spaces, linear maps, tensor products, coalgebras and algebras, etc., will be taken over a base field k. Let V be a vector space. This paper is concerned with the relation between the “cofree Lie coalgebra on I” and the “cofree covered Lie coalgebra on V.” A “Lie coalgebra” will be defined as a coalgebra that satisfies the “co-anticommutativity” and “co-Jacobi” Lie coidentities. This is analogous to the definition of a Lie algebra as an algebra that satisfies the anticommutativity and Jacobi identities. It follows from the Poincare-Birkhoff-Witt Theorem that every Lie algebra arises as a subalgebra of an A -, where A ~ is the Lie algebra associated to an associative algebra A. A “covered Lie coalgebra” will be defined as the quotient of some C-, where Cis the Lie coalgebra associated to an associative ( = coassociative) coalgebra C. Every covered Lie coalgebra satisfies the Lie coidentities, but, as was shown by Michaelis [Mill, there exist Lie coalgebras that do not arise as the quotient of any C-. Thus, covered Lie coalgebras are a specific type of Lie coalgebra. This situation for Lie coalgebras and covered Lie coalgebras is somewhat parallel to the situation for Jordan algebras and special Jordan algebras. (We recall that a Jordan algebra is an algebra that satisfies the Jordan identities and a special Jordan algebra is an algebra which arises as a subalgebra of an A +, A an associative algebra. Every special Jordan algebra is a Jordan algebra, that is, satisfies the identities, but there exist Jordan algebras which are not special.) For every vector space V one can construct the “cofree Lie coalgebra” and the “cofree covered Lie coalgebra” on V. Our main result (Theorem 1) will show that the canonical injective map from the cofree 431 0021-8693/88 $3.00