This paper aims to show that a certain moduli space T, which arises from the so-called Dwork family of Calabi-Yau n-folds, carries a special complex Lie algebra containing a copy of sl 2 (C). In order to achieve this goal, we introduce an algebraic group G acting from the right on T and describe its Lie algebra Lie(G). We observe that Lie(G) is isomorphic to a Lie subalgebra of the space of the vector fields on T. In this way, it turns out that Lie(G) and the modular vector field R generate another Lie algebra G, called AMSY-Lie algebra, satisfying dim(G) = dim(T). We find a copy of sl 2 (C) containing R as a Lie subalgebra of G. The proofs are based on an algebraic method calling "Gauss-Manin connection in disguise". Some explicit examples for n = 1, 2, 3, 4 are stated as well.

For any positive integer n, we consider a modular vector field R on a moduli space T of Calabi-Yau n-folds arising from the Dwork family enhanced with a certain basis of the n-th algebraic de Rham cohomology. The components of a particular solution of R, which are provided with definite weights, are called Calabi-Yau modular forms. Using R we introduce a derivation D and the Ramanujan-Serre type derivation ∂ on the space of Calabi-Yau modular forms. We show that they are degree 2 differential operators and there exists a proper subspace M 2 of the space of Calabi-Yau modular forms which is closed under ∂. Employing the derivation D, we define the Rankin-Cohen brackets for Calabi-Yau modular forms and prove that the subspace generated by the positive weight elements of M 2 is closed under the Rankin-Cohen brackets.

In this paper we obtain an ordinary differential equation H from a Picard-Fuchs equation associated with a nowhere vanishing holomorphic n-form. We work on a moduli space T constructed from a Calabi-Yau n-fold W together with a basis of the middle complex de Rham cohomology of W . We verify the existence of a unique vector field H on T such that its composition with the Gauss-Manin connection satisfies certain properties. The ordinary differential equation given by H is a generalization of differential equations introduced by Darboux, Halphen and Ramanujan.

Abstract. Many distinct problems give birth to Darboux-Halphen system of differential equations and here we review some of them. The first is the classical problem presented by Darboux and later solved by Halphen concerning finding infinite number of double orthogonal surfaces in R 3 . The second is a problem in general relativity about gravitational instanton in Bianchi IX metric space. The third problem stems from the new take on the moduli of enhanced elliptic curves called Gauss-Manin connection in disguise developed by one of the authors and finally in the last problem Darboux-Halphen system emerges from the associative algebra on the tangent space of a Frobenius manifold.

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