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Difference between revisions of "Akivis algebra"

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A [[Vector space|vector space]] over a [[Field|field]] with an anti-symmetric bilinear multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104501.png" /> and a multilinear ternary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104502.png" /> which are linked by the so-called Akivis condition, defined as follows [[#References|[a4]]], [[#References|[a5]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104503.png" /> denote the group of all six permutations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104504.png" /> the subgroup of all three cyclic permutations of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104505.png" />. Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104506.png" />. The Akivis condition reads:
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A [[Vector space|vector space]] over a [[Field|field]] with an anti-symmetric bilinear multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104501.png" /> and a multilinear [[ternary operation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104502.png" /> which are linked by the so-called Akivis condition, defined as follows [[#References|[a4]]], [[#References|[a5]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104503.png" /> denote the group of all six permutations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104504.png" /> the subgroup of all three cyclic permutations of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104505.png" />. Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104506.png" />. The Akivis condition reads:
  
 
<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/a/a110/a110450/a1104507.png" /></td> </tr></table>
 
<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/a/a110/a110450/a1104507.png" /></td> </tr></table>

Revision as of 18:02, 21 November 2014

A vector space over a field with an anti-symmetric bilinear multiplication and a multilinear ternary operation which are linked by the so-called Akivis condition, defined as follows [a4], [a5]. Let denote the group of all six permutations and the subgroup of all three cyclic permutations of the set . Define . The Akivis condition reads:

The specialization yields a Lie algebra. If is an arbitrary non-associative algebra over a field with a binary bilinear multiplication (cf. also Non-associative rings and algebras), then and define an Akivis algebra on . 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 -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
How to Cite This Entry:
Akivis algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Akivis_algebra&oldid=34698
This article was adapted from an original article by K.H. Hofmann (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article