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Complete classification of isotopically invariant varieties of analytic loops defined by regular identities of length four. (English. Russian original) Zbl 1391.53014

Russ. Math. 61, No. 3, 58-66 (2017); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2017, No. 3, 68-77 (2017).
A “quasi group” is a generalization of a group in the sense that it is a set with a multiplication where division is always possible. The multiplication need not be associative. A “loop” is a quasi-group with an “identity element”. A “Moufang loop” is a smooth manifold \(S\) with a multiplication that is a smooth morphism of manifolds, with the property that the underlying set with multiplication is a loop. A Moufang loop has an associated “Malcev algebra” and passing from the Moufang loop to the malcev algebra is a generalization of the passage from a Lie group to its associated Lie algebra. As in the theory of Lie groups and Lie algebras one may in some restricted situations recover the Moufang loop from its Malcev algebra.
The paper under review claims to give a complete classification of isotopically invariant varieties of analytic loops defined by regular identities of length four. The main results claimed in the paper are:
Theorem 1. If \(T\) is a regular identity in four variables then it is universal in the variety of loops.
Theorem 2. If \(T\) is an identity in one variable then it is universal in the variety of analytic loops \(H\) defined by the derived identity of the monoassociativity identity.
Theorem 3. If \(T\) is a regular identity in four variables (an identity of type \(B\)) then it is universal in one of the following \(I\)-invariant varieties of analytic loops: \(R\), \(B(l)\), \(B(r)\), \(M\).
The main theorem of the paper is the following:
Theorem 4. There are only twenty six regular identities in two variables. Among the \(I\)-invariant varieties of analytic loops defined by them, six varieties coincide with \(B(l)\), two varieties coincide with \(E\), one variety belongs to \(B(r)\), six varieties belong to \(M\), four varieties are contained in \(B(m)\) and \(E\).
Reviewer’s remark: If you are a non-expert the paper is difficult to read. It contains few proofs and few precise definitions.

MSC:

53A60 Differential geometry of webs
14C21 Pencils, nets, webs in algebraic geometry
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