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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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{{MSC|17A40}}
  
<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>
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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:
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$$
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\sum_{\sigma \in S_3} \mathrm{sgn}(\sigma) \left\langle x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}\right\rangle = J(x_1,x_2,x_3) \ .
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$$
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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]]].
  
The specialization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104508.png" /> yields a [[Lie algebra|Lie algebra]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a1104509.png" /> is an arbitrary non-associative algebra over a field with a binary bilinear multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a11045010.png" /> (cf. also [[Non-associative rings and algebras|Non-associative rings and algebras]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a11045011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a11045012.png" /> define an Akivis algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a11045013.png" />. The tangent algebra of a local analytic loop (cf. [[Loop, analytic|Loop, analytic]]) is always an Akivis algebra. This generalizes the facts that the tangent algebra of a local [[Lie group|Lie group]] (cf. also [[Lie group, local|Lie group, local]]) is a [[Lie algebra|Lie algebra]] and that the tangent algebra of a local [[Moufang loop|Moufang loop]] is a [[Mal'tsev algebra|Mal'tsev algebra]]. Analytic or differentiable quasi-groups (cf. [[Quasi-group|Quasi-group]]) and loops arise in the study of the geometry of webs (cf. [[Web|Web]]), [[#References|[a2]]], [[#References|[a3]]], [[#References|[a5]]].
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====References====
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<table>
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<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>
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<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>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  V.V. Goldberg,  "Theory of multicodimensional $(n+1)$-webs" , Kluwer Acad. Publ. (1988)</TD></TR>
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<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>
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<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>
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<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>
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</table>
  
====References====
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{{TEX|done}}
<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110450/a11045014.png" />-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>
 

Latest revision as of 21:20, 18 December 2015

2020 Mathematics Subject Classification: Primary: 17A40 [MSN][ZBL]

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
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