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
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 ). A power-associative analytic loop (cf. Algebra with associative powers) has canonical coordinates of the first kind . 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 .
The most fully studied are analytic Moufang loops (cf. Moufang loop). The tangent algebra of an analytic Moufang loop satisfies the identities
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 , ). 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 ).
|||A.I. Mal'tsev, "Analytic loops" Mat. Sb. , 36 : 3 (1955) pp. 569–578 (In Russian)|
|||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|
|||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|
|[a1]||O. Chein (ed.) H. Pflugfelder (ed.) J.D.H. Smith (ed.) , Theory and application of quasigroups and loops , Heldermann (1989)|
Loop, analytic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Loop,_analytic&oldid=33034