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Let G be a classical group of dimension d and let a = (a 1 , ... , a d) be differential indeterminates over a differential field F of characteristic zero with algebraically closed field of constants C. Further let A (a) be a generic element in the Lie algebra g (F 〈 a 〉) of G obtained from parametrizing a basis of g with the indeterminates a. It is known (cf. [5]) that the differential Galois group of y ′ = A (a) y over F 〈 a 〉 is G (C) . In this paper we construct a differential field extension L of F 〈 a 〉 such that the field of constants of L is C, the differential Galois group of y ′ = A (a) y over L is still the full group G (C) and A (a) is gauge equivalent over L to a matrix in normal form which we introduced in [13]. We also consider specializations of the coefficients of A (a) . [ABSTRACT FROM AUTHOR]

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action αG of a finite group G on an algebra S such that S is an α G -partial Galois extension of S α G and a normal subgroup H of G, we prove that α G induces a unital partial action α G / H of G/H on the subalgebra of invariants S α H of S such that S α H is an α G / H -partial Galois extension of S α G . Second, assuming that G is abelian, we construct a commutative inverse semigroup T par (G , R) , whose elements are equivalence classes of α G -partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between T par (G , R) / ρ and T(G, A), where ρ is a congruence on T par (G , R) and T(G, A) is the classical Harrison group of the G-isomorphism classes of the abelian extensions of a commutative algebra A. It is shown that the study of T par (G , R) reduces to the case where G is cyclic. The set of idempotents of T par (G , R) is also investigated. [ABSTRACT FROM AUTHOR]