We study systematically the natural generalization of Schreier's extension theory to obtain proper loops and show that this construction gives a rich family of examples of loops in all traditional common, important loop classes. Keywords: extension of loops, non-associative extension of groups, weak associativity properties of extensions, central extensions MSC 2000 : 20N05
A central piece of classical Lie theory is the fact that with each local Lie group, a Lie algebra is associated as tangent object at the origin, and that, conversely and more importantly, every Lie algebra determines a local Lie group whose tangent algebra it is. Up to equivalence of local groups, this correspondence is bijective.Attempts at the development of a Lie theory for analytical loops have not been entirely satisfactory in this direction, since they relied more or less on certain associativity assumptions. Here we associate with an arbitrary local analytical loop a unique tangent algebra with a ternary multiplication in addition to the standard binary one, and we call this algebra an Akivis algebra. Our main objective is to show that, conversely, for every Akivis algebra there exist many inequivalent local analytical loops with the given Akivis algebra as tangent algebra. We shall give a good idea about the degree of non-uniqueness. It is curious to note that, on account of this non-uniqueness, the construction is more elementary than in the case of analytical groups. where H n {X,Y) are the homogeneous polynomials of degree w, being defined as the linear combinations of the monomials of degree n occurring in the series, then it turns out that H n (X,Y) is always contained in the Lie algebra L [X, Y] A loop is a group except that one allows the multiplication to be non-associative. Specifically: A loop is a set G with a distinguished element e and three binary operations (x, y) *-> xy, x/y, y\x: G X G -> G such that xe = ex = x and y(y\x) = (x/y)y = x for all x, y e G. Topological and analytical loops are easily defined, and with the usual circumspection one defines local analytical loops (see §3 below or [2]). In this paper we prove Lie's fundamental theorems for local analytical loops.Why bother? There are several reasons. Firstly, Lie loops have been considered in the literature at least since that paper by A. I. Malcev [20] which is considered the start of the theory. The area of Moufang Lie loops is in fact a highly developed theory for which most results of classical Lie group theory have been verified mostly through the work of Kuzmin and Kerdman [14,15,16,17] and through Sagle [22,23,24]. In his extensive study of webs in differential geometry, Akivis and his school [1 through 7, 21] contributed substantially to the general foundations of Lie loop theory. However, Lie's Third fundamental theorem was never established (see [3]). Holmes and Sagle studied local power associative Lie loops and dealt with the question to which extent the Campbell-Hausdorff formalism could be brought to bear on the theory [13]. A similar discussion is to be found in a paper by Akivis [1].However, apart from the applications to Lie loop theory we feel that a discussion of Lie's fundamental theorems in the absence of associativity LIE'S THEOREMS FOR LOOPS 303 sheds new light on the classical associative case. In fact, before we can deal with the non-associative case we first have to find the appropriate generali...
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