Lie’s fundamental theorems for local analytical loops

Abstract

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...

The Akivis algebra of a homogeneous loop

“…This local multiplication was considered by Sagle and Schumi [6]. The implicit function theorem implies that it defines indeed a local Lie loop (see [2,3]). We do not claim at this point that the operation extends to a global one on the entire homogeneous space making it into an analytical loop.…”

“…(vi) £ sgn (g) • (X gW , X g (2) , Xg ( where 5 3 is the group of permutations on the set {1, 2,3}. The identity (vi) was formulated by Akivis [1], and in [2,3] we have called a vector space an Akivis algebra if it carries an anticommutative bilinear and a trilinear operation which are linked by the identity (vi). If M is a Lie group, then the Akivis algebra L(M) reduces to the traditional Lie algebra.…”

“…If M is a Lie group, then the Akivis algebra L(M) reduces to the traditional Lie algebra. We have shown in [2] that for every finite dimensional Akivis algebra over IR there is at least one local Lie loop which has the given Akivis algebra as tangent algebra.…”

On (n+1)-subwebs of an (n+1)-web and local algebras associated with them

“…It is known that an algebra structure is associated with any web W(n + 1, n, r) (see [19] or [16]). This structure consists of (~) A-algebras (in [17] they were called Akivis algebras and originally in [3] they were called W-algebras) and (~) C-algebras (in [19] and [16] they are called comtrans algebras).…”

“…The transversal (n + 1)-subwebs were studied for n = 2, s = 1 in [1] and for n > 2, s = 1 in [9] (see also [10] or [16]). It was proved in these papers that n-dimensional (n+l)-subwebs, n => 2, are induced on transversal geodesic surfaces of the (n + 1)-web W. In the paper [6] transversal 3-subwebs of a 3-web W(3, 2, r) were studied for s = 1,... , r -1, the relationship of the Akivis algebras defined by the coordinate loops of the 3-web and its transversal 3-subwebs (see [3], [6] or [17]) was established, and isoclinic and Grassmann 3-webs were characterized in terms of these algebras.…”