View source for Loop, analytic

An [[Analytic manifold|analytic manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l0608401.png" /> endowed with the structure of a [[Loop|loop]] whose basic operations (multiplication, left and right division) are analytic mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l0608402.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l0608403.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l0608404.png" /> is the identity of the loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l0608405.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l0608406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l0608407.png" /> are analytic paths starting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l0608408.png" /> and having tangent vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l0608409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084010.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084011.png" />, then the tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084012.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084013.png" /> to the path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084014.png" />, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084015.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084016.png" /> stands for right division, is a bilinear function of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084018.png" />. The tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084019.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084020.png" /> with the operation of multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084021.png" /> is called the tangent algebra of the loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084022.png" />. In some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084023.png" /> of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084024.png" /> the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084025.png" /> are said to be canonical of the first kind if for any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084026.png" /> the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084027.png" /> is a local one-parameter subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084028.png" /> with tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084029.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084030.png" /> (see [[#References|[1]]]). A power-associative analytic loop (cf. [[Algebra with associative powers|Algebra with associative powers]]) has canonical coordinates of the first kind [[#References|[2]]]. In this case the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084031.png" />, defined for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084032.png" />, makes it possible to identify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084033.png" /> with a neighbourhood of the origin in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084034.png" /> and to endow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084035.png" /> with the structure of a local analytic loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084036.png" />. If an analytic loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084037.png" /> is alternative, that is, if any two elements of it generate a subgroup, then the tangent algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084038.png" /> is a [[Binary Lie algebra|binary Lie algebra]], and the multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084039.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084040.png" /> can be expressed by the [[Campbell–Hausdorff formula|Campbell–Hausdorff formula]]. Any finite-dimensional binary Lie algebra over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084041.png" /> is the tangent algebra of one and only one (up to local isomorphisms) local alternative analytic loop [[#References|[1]]].

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084042.png" /></td> </tr></table>

where

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such algebras are called Mal'tsev algebras. Conversely, any finite-dimensional Mal'tsev algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084044.png" /> is the tangent algebra of a simply-connected analytic Moufang loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084045.png" />, defined uniquely up to an isomorphism (see [[#References|[2]]], [[#References|[3]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084046.png" /> is a connected analytic Moufang loop with the same tangent algebra, and hence is locally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084047.png" />, then there is an epimorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084048.png" /> whose kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084049.png" /> is a discrete normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084050.png" />; the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084051.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084052.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084053.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084054.png" /> is a local homomorphism of a simply-connected analytic Moufang loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084055.png" /> into a connected analytic Moufang loop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084057.png" /> can be uniquely extended to a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084058.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084059.png" />. The space of a simply-connected analytic Moufang loop with solvable Mal'tsev tangent algebra is analytically isomorphic to the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060840/l06084060.png" /> (see [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,   "Analytic loops"  ''Mat. Sb.'' , '''36''' :  3  (1955)  pp. 569–578  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Chein (ed.)  H. Pflugfelder (ed.)  J.D.H. Smith (ed.) , ''Theory and application of quasigroups and loops'' , Heldermann  (1989)</TD></TR></table>