Akivis algebra

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 [a4], [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), [a2], [a3], [a5].

References

[a1]	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
[a2]	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
[a3]	V.V. Goldberg, "Theory of multicodimensional $(n+1)$-webs" , Kluwer Acad. Publ. (1988)
[a4]	K.H. Hofmann, K. Strambach, "The Akivis algebra of a homogeneous loop" Mathematika , 33 (1986) pp. 87–95
[a5]	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
[a6]	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