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Polynomial identities of RA and RA2 loop algebras. (English) Zbl 1132.20043

RA loops are loops whose loop rings are alternative rings, and RA2 loops are those with alternative loop rings in characteristic two. Both RA and RA2 loops are Moufang loops.
By the so-called Chein process a non-associative Moufang loop \(L\) is constructed starting with a non-Abelian group \(G\) and an involution \(^*\) of \(G\) with \(g_0^*=g_0\), where \(g_0\) is a central element of \(G\). This loop is denoted by \(M(G,^*,g_0)\).
RA loops are loops of type \(M(G,^*,g_0)\). In contrary to that, not all RA2 loops are of this type.
Let \(F\) be an algebraically closed field of characteristic zero. It is proved that the loop algebra \(FL\) of any RA loop \(L\) is in the variety generated by the split Cayley-Dickson algebra \(\mathcal Z_F\) over \(F\).
For RA2 loops things are more complicated: Assume \(L=M(\text{Dih}(A),^{-1},g_0)\), where \(\text{Dih}(A)\) is the dihedral group \(A\rtimes C_2\), \(A\) Abelian, \(C_2\) cyclic of order 2. Then the loop algebra \(FL\) is in the variety generated by the algebra \(\mathcal A_3\) which is a non-commutative simple component of a certain RA2 loop of order 16.
If the RA2 loop \(L\) is of type \(M(G,^{-1},g_0)\) where \(G\) is a non-Abelian group of exponent 4 having exactly 2 squares this does not hold: It is shown that there cannot exist a finite-dimensional \(F\)-algebra \(\mathcal D\) such that \(FL\) is in the variety generated by \(\mathcal D\) for all \(G\).

MSC:

20N05 Loops, quasigroups
17D05 Alternative rings
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References:

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