Primitive elements of free nonassociative algebras

Михалев, А. В.; Mikhalëv, A. V.; Chepovskiy, A. A.; Champagnier, K.

doi:10.1007/s10958-008-9269-y

Cited by 4 publications

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“…(2) is much wider. Indeed, consider the elements u 3,3 (x, y) = (ad x) 3 (y) + (x)(Ad y) 3 and u 4,4 (x, y) = (ad x) 4 (y) + (x)(Ad y) 4 . It was proved in [6] that the elements u k,l (x, y) = (ad x) k (y) + (x)(Ad y) l are almost primitive in algebra L(x, y) when k, l ≥ 2, k = l. This is equivalent to the system of equations…”

Section: Theorem 1 (Criterion For Almost Primitivity Of a Homogeneousmentioning

confidence: 99%

Almost Primitive Elements of Free lie Algebras of Small Ranks

Klimakov

2014

J Math Sci

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Let K be a field, X = {x1, . . . , xn}, and let L(X) be the free Lie algebra over K with the set X of free generators. A. G. Kurosh proved that subalgebras of free nonassociative algebras are free, A. I. Shirshov proved that subalgebras of free Lie algebras are free. A subset M of nonzero elements of the free Lie algebra L(X) is said to be primitive if there is a set Y of free generators of L(X), L(X) = L(Y ), such that M ⊆ Y (in this case we have |Y | = |X| = n). Matrix criteria for a subset of elements of free Lie algebras to be primitive and algorithms to construct complements of primitive subsets of elements with respect to sets of free generators have been constructed. A nonzero element u of the free Lie algebra L(X) is said to be almost primitive if u is not a primitive element of the algebra L(X), but u is a primitive element of any proper subalgebra of L(X) that contains it. A series of almost primitive elements of free Lie algebras has been constructed. In this paper, for free Lie algebras of rank 2 criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed.

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Section: Theorem 1 (Criterion For Almost Primitivity Of a Homogeneousmentioning

confidence: 99%