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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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