Loop, analytic

An analytic manifold endowed with the structure of a loop whose basic operations (multiplication, left and right division) are analytic mappings of into . If is the identity of the loop , and and are analytic paths starting from and having tangent vectors and at , then the tangent vector at to the path , where

where stands for right division, is a bilinear function of the vectors and . The tangent space at with the operation of multiplication is called the tangent algebra of the loop . In some neighbourhood of the element the coordinates are said to be canonical of the first kind if for any vector the curve is a local one-parameter subgroup with tangent vector at (see [1]). A power-associative analytic loop (cf. Algebra with associative powers) has canonical coordinates of the first kind [2]. In this case the mapping , defined for sufficiently small , makes it possible to identify with a neighbourhood of the origin in and to endow with the structure of a local analytic loop . If an analytic loop is alternative, that is, if any two elements of it generate a subgroup, then the tangent algebra is a binary Lie algebra, and the multiplication in can be expressed by the Campbell–Hausdorff formula. Any finite-dimensional binary Lie algebra over the field is the tangent algebra of one and only one (up to local isomorphisms) local alternative analytic loop [1].

The most fully studied are analytic Moufang loops (cf. Moufang loop). The tangent algebra of an analytic Moufang loop satisfies the identities

where

such algebras are called Mal'tsev algebras. Conversely, any finite-dimensional Mal'tsev algebra over is the tangent algebra of a simply-connected analytic Moufang loop , defined uniquely up to an isomorphism (see [2], [3]). If is a connected analytic Moufang loop with the same tangent algebra, and hence is locally isomorphic to , then there is an epimorphism whose kernel is a discrete normal subgroup of ; the fundamental group of the space is isomorphic to . If is a local homomorphism of a simply-connected analytic Moufang loop into a connected analytic Moufang loop , then can be uniquely extended to a homomorphism of into . The space of a simply-connected analytic Moufang loop with solvable Mal'tsev tangent algebra is analytically isomorphic to the Euclidean space (see [3]).

References

[1]	A.I. Mal'tsev, "Analytic loops" Mat. Sb. , 36 : 3 (1955) pp. 569–578 (In Russian)
[2]	E.N. Kuz'min, "On the relation between Mal'tsev algebras and analytic Moufang loops" Algebra and Logic , 10 : 1 (1971) pp. 1–14 Algebra i Logika , 10 : 1 (1971) pp. 3–22
[3]	F.S. Kerdman, "On global analytic Moufang loops" Soviet Math. Dokl. , 20 (1979) pp. 1297–1300 Dokl. Akad. Nauk SSSR , 249 : 3 (1979) pp. 533–536

References

[a1]	O. Chein (ed.) H. Pflugfelder (ed.) J.D.H. Smith (ed.) , Theory and application of quasigroups and loops , Heldermann (1989)