View source for Akivis algebra

{{MSC|17A40}}

A [[vector space]] over a [[field]] with an anti-symmetric bilinear multiplication $(x,y) \mapsto [x,y]$ and a multilinear [[ternary operation]] $(x,y,z) \mapsto \langle x,y,z\rangle$ which are linked by the so-called Akivis condition, defined as follows [[#References|[a4]]], [[#References|[a5]]]. Let $S_3$ denote the group of all six permutations and $A_3$ the subgroup of all three cyclic permutations of the set $\{1,2,3\}$. Define $J(x,y,z) = \sum_{\sigma \in A_3} \left[{\left[x_{\sigma(1)},x_{\sigma(2)}\right],x_{\sigma(3)}}\right]$. The Akivis condition reads:
$$
\sum_{\sigma \in S_3} \mathrm{sgn}(\sigma) \left\langle x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}\right\rangle = J(x_1,x_2,x_3) \ .
$$
The specialization $\langle x,y,z \rangle \equiv 0$ yields a [[Lie algebra]]. If $A$ is an arbitrary non-associative algebra over a field with a binary bilinear multiplication $(x,y) \mapsto x \cdot y$ (cf. also [[Non-associative rings and algebras]]), then $[x,y] = x \cdot y - y \cdot x$ and $\langle x,y,z \rangle = (x \cdot y) \cdot z - x \cdot (y \cdot z)$ define an Akivis algebra on $A$. The tangent algebra of a local analytic loop (cf. [[Loop, analytic]]) is always an Akivis algebra. This generalizes the facts that the tangent algebra of a local [[Lie group]] (cf. also [[Lie group, local]]) is a [[Lie algebra]] and that the tangent algebra of a local [[Moufang loop]] is a [[Mal'tsev algebra]]. Analytic or differentiable quasi-groups (cf. [[Quasi-group]]) and loops arise in the study of the geometry of webs (cf. [[Web]]), [[#References|[a2]]], [[#References|[a3]]], [[#References|[a5]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M.A. Akivis,   "The canonical expansions of the equations of a local analytic quasigroup"  ''Soviet Math. Dokl.'' , '''10'''  (1969)  pp. 1200–1203  ''Dokl. Akad. Nauk SSSR'' , '''188'''  (1969)  pp. 967–970</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  V.V. Goldberg,   "Local differentiable quasigroups and webs"  O. Chein (ed.)  H.O. Pflugfelder (ed.)  J.D.H. Smith (ed.) , ''Quasigroups and Loops - Theory and Applications'' , Heldermann  (1990)  pp. 263–311</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  V.V. Goldberg,   "Theory of multicodimensional $(n+1)$-webs" , Kluwer Acad. Publ.  (1988)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  K.H. Hofmann,   K. Strambach,   "The Akivis algebra of a homogeneous loop"  ''Mathematika'' , '''33'''  (1986)  pp. 87–95</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  K.H. Hofmann,   K. Strambach,   "Topological and analytic loops"  O. Chein (ed.)  H.O. Pflugfelder (ed.)  J.D.H. Smith (ed.) , ''Quasigroups and Loops - Theory and Applications'' , Heldermann  (1990)  pp. 205–262</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  P.O. Miheev,   L.V. Sabinin,   "Quasigroups and differential geometry"  O. Chein (ed.)  H.O. Pflugfelder (ed.)  J.D.H. Smith (ed.) , ''Quasigroups and Loops - Theory and Applications'' , Heldermann  (1990)  pp. 357–430</TD></TR>
</table>

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