Every monomorphism of the Lie algebra of triangular polynomial derivations is an automorphism

Bavula, V. V.

doi:10.1016/j.crma.2012.06.001

Cited by 11 publications

(9 citation statements)

References 5 publications

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“…By Lemma 3.1. (3,4) and Lemma 3.10. (1), if x α > x β (where α, β ∈ N n ) then (i) ∂ k * x α > ∂ k * x β for all k = 1, .…”

Section: Ann Unmentioning

confidence: 94%

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Lie algebras of triangular polynomial derivations and an isomorphism criterion for their Lie factor algebras

Bavula¹

2013

Izv. Math.

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The Lie algebras un (n ≥ 2) of triangular polynomial derivations, their injective limit u∞ and the completion u∞ are studied in detail. The ideals of un are classified, all of them are characteristic ideals. Using the classification of ideals, an explicit criterion is given for the Lie factor algebras of un and um to be isomorphic. For (Lie) algebras (and their modules) two new dimensions are introduced: the central dimension c.dim and the uniserial dimension u.dim. It is shown that c.dim(un) = u.dim(un) = ω n−1 + ω n−2 + · · · + ω + 1 for all n ≥ 2 where ω is the first infinite ordinal. Similar results are proved for the Lie algebras u∞ and u∞. In particular, u.dim(u∞) = ω ω and c.dim(u∞) = 0.Key Words: Lie algebra, triangular polynomial derivations, automorphism, isomorphism problem, the derived and upper central series, locally nilpotent derivation, locally nilpotent and locally finite dimensional Lie algebra.Mathematics subject classification 2010: 17B66, 17B40, 17B65, 17B30, 17B35.

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“…By Lemma 3.1. (3,4) and Lemma 3.10. (1), if x α > x β (where α, β ∈ N n ) then (i) ∂ k * x α > ∂ k * x β for all k = 1, .…”

Section: Ann Unmentioning

confidence: 94%

“…A new dimension for algebras and modules is introduced -the uniserial dimension (Section 4) -that turned out to be a very useful tool in studying non-Noetherian Lie algebras, their ideals and automorphisms, [3,4,5].…”

Section: Introductionmentioning

confidence: 99%

Lie algebras of triangular polynomial derivations and an isomorphism criterion for their Lie factor algebras

Bavula¹

2013

Izv. Math.

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“…[8]. He also proved that every monomorphism of the Lie algebra of triangular polynomial derivations is an automorphism [9] (an analog of Dixmier's conjecture).…”

Section: Introductionmentioning

confidence: 97%

The Jacobian Conjecture, Together with Specht and Burnside-Type Problems

Kanel-Belov

Bokut

Rowen

et al. 2014

Springer Proceedings in Mathematics &Amp; Statistics

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Dedicated to the memory of A.V. Yagzhev Abstract. We explore an approach to the celebrated Jacobian Conjecture by means of identities of algebras, initiated by the brilliant deceased mathematician, Alexander Vladimirovich Yagzhev , only partially published. This approach also indicates some very close connections between mathematical physics, universal algebra and automorphisms of polynomial algebras.

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“…The groups Aut Lie (u n ) and Aut Lie (D n ) where found in [3] and [4] respectively. The Lie algebras u n have been studied in great detail in [1] and [2]. In particular, in [1] it was proved that every monomorphism of the Lie algebra u n is an automorphism but this is not true for epimorphisms.…”

Section: Introductionmentioning

confidence: 99%