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Combinatorial properties of free algebras of Schreier varieties. (English) Zbl 1107.17300

Giambruno, Antonio (ed.) et al., Polynomial identities and combinatorial methods, Pantelleria, Italy. New York, NY: Marcel Dekker (ISBN 0-8247-4051-3/pbk). Lect. Notes Pure Appl. Math. 235, 47-99 (2003).
A variety of algebras over a field satisfies the Schreier property, if any subalgebra of a free algebra in the variety is free again. Typical examples of such algebras are the free Lie algebras and color Lie superalgebras, Lie \(p\)-algebras and color Lie \(p\)-superalgebras, the free commutative and anticommutative (nonassociative) algebras and the absolutely free algebras. All these algebras have the remarkable property that their automorphisms are tame, i.e. are products of elementary automorphisms. The purpose of the paper under review is to survey results concerning automorphic orbits of elements of the free algebras of the main types of Schreier varieties, on Schreier techniques, on applications of free differential calculus, on primitive, test, and generalized primitive elements of free algebras. The authors consider also similar problems for some free algebras of varieties which are not Schreier (“ordinary” polynomial algebras, free associative algebras, free Leibniz algebras) and for free groups, as a source of motivation or to show some analogy and differences. A long list of references is given.
For the entire collection see [Zbl 1027.00013].

MSC:

17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
17A36 Automorphisms, derivations, other operators (nonassociative rings and algebras)
17A50 Free nonassociative algebras
17B01 Identities, free Lie (super)algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
17B75 Color Lie (super)algebras
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