Loop, analytic

An analytic manifold $M$ endowed with the structure of a loop whose basic operations (multiplication, left and right division) are analytic mappings of $M\times M$ into $M$. If $e$ is the identity of the loop $M$, and $g(t)$ and $h(t)$ are analytic paths starting from $e$ and having tangent vectors $a$ and $b$ at $e$, then the tangent vector $c=ab$ at $e$ to the path $k(t)$, where

$$k(t^2)=(g(t)h(t))/(h(t)g(t)),$$

where $/$ stands for right division, is a bilinear function of the vectors $a$ and $b$. The tangent space $T(M)$ at $e$ with the operation of multiplication $c=ab$ is called the tangent algebra of the loop $M$. In some neighbourhood $U$ of the element $e=(0,\dots,0)$ the coordinates $(x^1,\dots,x^n)$ are said to be canonical of the first kind if for any vector $a=(a^1,\dots,a^n)$ the curve $x(t)=(a^1t,\dots,a^nt)$ is a local one-parameter subgroup $(|t|\leq\epsilon)$ with tangent vector $a$ at $e$ (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 $a\to x(1)$, defined for sufficiently small $a$, makes it possible to identify $U$ with a neighbourhood of the origin in $T(M)$ and to endow $T(M)$ with the structure of a local analytic loop $M_0$. If an analytic loop $M$ is alternative, that is, if any two elements of it generate a subgroup, then the tangent algebra $T(M)$ is a binary Lie algebra, and the multiplication $(x,y)\to x\circ y$ in $M_0$ can be expressed by the Campbell–Hausdorff formula. Any finite-dimensional binary Lie algebra over the field $\mathbf R$ 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

$$x^2=0,\quad J(x,y,xz)=J(x,y,z)x,$$

where

$$J(x,y,z)=(xy)z+(yz)x+(zx)y;$$

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