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Multilinear algebras and Lie’s theorem for formal $$n$$-loops. (English) Zbl 0627.22003
An $$(n+1)$$-web $$W(n+1,n,r)$$ is coordinatized by a local real analytic $$n$$-ary loop, whose Taylor expansion furnishes an $$r$$-dimensional formal real $$n$$-ary loop. For $$n=2$$, the connection between formal groups and Lie algebras or formal Moufang loops and Mal’cev algebras was extended by M. A. Akivis [Sib. Math. Zh. 17, 5–11 (1976; Zbl 0337.53018)], K. H. Hofmann and K. Strambach [Pac. J. Math. 123, 301–327 (1986; Zbl 0596.22002)] to a connection between formal binary loops and Akivis algebras $$(A,[\cdot,\cdot], (\cdot,\cdot,\cdot))$$, with a bilinear commutator $$[x,y]=xy-yx+0(3)$$ and trilinear associator $$(x,y,z)=(xy)z-x(yz)+0(4)$$. V. V. Goldberg has raised the problem of finding analogous results for $$n>2.$$
Certainly, a trilinear product $$xyz$$ on a space $$A$$ furnishes a trilinear commutator $$[x,y,z]=xyz-yxz$$ and a trilinear translator $$\langle x,y,z\rangle=xyz- yzx$$. The commutator is left alternative $$[x,y,z]+[y,x,z]=0$$, the translator satisfies the Jacobi identity $$\langle x,y,z\rangle+\langle y,z,x\rangle+\langle z,x,y\rangle=0$$, and together the commutator and translator satisfy the comtrans identity $$[x,y,z]+[z,y,x]=\langle x,y,z\rangle+\langle z,y,x\rangle$$. An algebra $$(A,[\cdot,\cdot,\cdot], \langle\cdot,\cdot,\cdot\rangle)$$ satisfying these identities is called a comtrans algebra.
The peculiar difficulty of formal $$n$$-ary loops for $$n>2$$ is that multilinear operations are not obtained directly as in the binary case. For example, a formal ternary loop $$F(x,y,z)$$ does not give a trilinear commutator $$F(x,y,z)-F(y,x,z)+0(4)$$. To obtain trilinear operations, the technique of masking is introduced. The masks are the formal binary loops $$F(0,y,z)$$, $$F(x,0,z)$$, and $$F(x,y,0)$$. Each has a corresponding Akivis algebra. A comtrans algebra is then obtained from the commutator and translator of the masked version $$M(x,y,z)=F(0,y,z)+F(x,0,z)+F(x,y,0)- F(x,y,z)$$ of $$F(x,y,z)$$. The full algebra structure corresponding to the formal ternary loop thus consists of a comtrans algebra and the three Akivis algebras of the masks. An analogue of Lie’s third fundamental theorem governs this correspondence. More generally, a formal $$n$$-ary loop or $$(n+1)$$-web $$W(n+1,n,r)$$ is desribed by $$\left( \begin{matrix} n\\ 2\end{matrix} \right)$$ Akivis algebras and $$\left( \begin{matrix} n\\ 3\end{matrix} \right)$$ comtrans algebras in the tangent space.
Reviewer: J. D. H. Smith

##### MSC:
 22A30 Other topological algebraic systems and their representations 17A40 Ternary compositions 53A60 Differential geometry of webs 20N05 Loops, quasigroups 14L05 Formal groups, $$p$$-divisible groups 17D10 Mal’tsev rings and algebras
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##### References:
 [1] M. A. Akivis, Local algebras of a multidimensional three-web, (Russian). Sibirsk. Mat. ?.17, 5-11 (1976). Translated in: Siberian Math. J.17, 3-8 (1976). [2] S. Bochner, Formal Lie groups. Ann. of Math.47, 192-201 (1946). · Zbl 0063.00488 · doi:10.2307/1969242 [3] O.Chein, H.Pflugfelder and J. D. H.Smith (eds.), Theory and Applications of Quasigroups and Loops. Berlin 1989. [4] C.Chevalley, Théorie des Groupes de Lie II. Paris 1951. · Zbl 0054.01303 [5] J.Dieudonné, Introduction to the Theory of Formal Groups. New York 1973. [6] V. V. Gol’dberg, An invariant characteristic of certain closure conditions in ternary quasigroups, (Russian). Sibirsk. Mat. ?.16, 29-43 (1975). Translated in: Siberian Math. J.16, 23-34 (1975). [7] M.Hazewinkel, Formal Groups and Applications. New York-London 1978. · Zbl 0454.14020 [8] K. H. Hofmann andK. Strambach, Lie’s fundamental theorems for local analytical loops. Pacific J. Math.123, 301-327 (1986). · Zbl 0596.22002 [9] E. N. Kuz’min, The connection between Mal’cev algebras and analytic Moufang loops, (Russian). Algebra i Logika10, 3-22 (1971). Translated in: Algebra and Logic10, 1-14 (1972). [10] S.Lie, Vorlesungen über continuierliche Gruppen. Leipzig 1893. · JFM 25.0623.02 [11] A. I. Mal’cev, Analytical loops, (Russian). Mat. Sb.36, 569-576 (1955). [12] C.Scheiderer, Gewebegeometrie 10.6 bis 16. 6. 1984. Tagungsbericht 27/1984, Mathematisches Forschungsinstitut Oberwolfach 1984.
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