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ABSTRACT
Using translates, a characterization for the dual coalgebra of any Lie algebra is given. This characterization is analogous to the well known characterization for the dual coalgebra of any associative algebra. For any commutative, associative algebra and finite set of commuting derivations
of
satisfying a certain additional hypothesis, the structure of the dual coalgebra of the Lie subalgebra
of
is determined. This generalizes a result NicholsCitation[1] proved for the case
. As an application, the family of dual coalgebras which corresponds to the family of infinite-dimensional Lie algebras of derivations of polynomials in several indeterminates is then given.
ACKNOWLEDGMENT
At this point I would like to express my sincere thanks to Richard BlockCitation[14]for suggesting that I rework my original manuscript in the manner ofCitation[1], and for showing me the present general version of Lemma 1, and Theorem 2 (I had discovered, with algebraically closed and using different techniques, the special case of Theorem 2 for
.)