Weitzenböck derivations of free metabelian Lie algebras

Dangovski, Rumen; Drensky, Vesselin; Findik, Sehmus

doi:10.1016/j.laa.2013.05.004

Cited by 10 publications

(25 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The map π sends v to the commutator ideal L ′ d /L ′′ d of the free metabelian Lie algebra L ′ d /L ′′ d . In [2] we have seen that δ(π(v)) = π(δ(v)). Now δ(π(w)) = δ(π(uv)) = δ(π(v)u) = δ(π(v))u+π(v)δ(u) = π(δ(v))u+π(v)δ(u) = π(δ(v)u) + π(vδ(u)) = π(δ(v)u + vδ(u)) = π(δ(vu)) = π(δ(w)) and this establishes (ii).…”

Section: Clearly This Implies Thatmentioning

confidence: 91%

“…Starting from the Hilbert series of the algebra F d = F d (M) and the sizes of the Jordan cells of the Weitzenböck derivation δ, there are many ways to compute the Hilbert series of the algebra of constants F δ d , see the comments in [1] and [2]. We shall sketch the way used in [2] in the Lie case. It is an improvement (given in [1]) of the method of Elliott [6] and its further development by McMahon [8], the so called partition analysis or Ω-calculus.…”

Section: Hilbert Seriesmentioning

confidence: 99%

“…[x 3 , x 1 ]u 3 v 1 −[x 3 , x 1 ]u 1 v 3 +[x 3 , x 2 ]u 1 v 2 −[x 3 , x 2 ]u 2 v 1 −[x 2 , x 1 ]u 3 v 2 +[x 2 , x 1 ]u 2 v 3of bidegree (4, 4) and the relations R 1(6,2) :c 1 f 6 = c 3 f 2 − c 2 f 1 , R 2 (7, 3) : c 2 f 6 = c 4 f 2 2 − c 1 (f 1 f 4 + f 2 f 5 ), R 3 (7, 3) : c 3 f 6 = c 4 f 1 f 2 + c 1 (f 1 f 5 + f 2 f 3 ), R 4 (6, 4) : c 4 f 6 = c 2 f 3 + c 3 f 5 + c 5 f 1 , R 5 (6, 4) : c 5 f 2 = c 2 f 5 + c 3 f 4 , R 6 (7, 5) : c 5 f 6 = c 1 (f 3 f 4 − f 2 5 ) + c 4 (f 1 f 4 + f 2 f 5 ),The above relations show that c j f 6 , j = 1, . .…”

mentioning

confidence: 99%

See 2 more Smart Citations

Weitzenböck derivations of free metabelian associative algebras

2017

Self Cite

View full text Add to dashboard Cite

By the classical theorem of Weitzenböck the algebra of constants K[X d ] δ of a nonzero locally nilpotent linear derivation δ of the polynomial algebra K[X d ] = K[x1, . . . , x d ] in several variables over a field K of characteristic 0 is finitely generated. As a noncommutative generalization one considers the algebra of constants F d (V) δ of a locally nilpotent linear derivation δ of a finitely generated relatively free algebra F d (V) in a variety V of unitary associative algebras over K. It is known that F d (V) δ is finitely generated if and only if V satisfies a polynomial identity which does not hold for the algebra U2(K) of 2 × 2 upper triangular matrices. Hence the free metabelian associative algebra F d = F d (M) = F d (N2A) = F d (var(U2(K))) is a crucial object to study. We show that the vector space of the constants (where δ acts on U d and V d in the same way as on X d . For small d, we calculate the Hilbert series of (F ′ d ) δ and find the generators of the K[U d , V d ] δ -module (F ′ d ) δ . This gives also an (infinite) set of generators of the algebra F δ d .Free metabelian associative algebras; algebras of constants; Weitzenböck derivations.[2010]16R10; 16S15; 16W20; 16W25; 13N15; 13A50.

show abstract

Section: Clearly This Implies Thatmentioning

confidence: 91%

Section: Hilbert Seriesmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Weitzenböck derivations of free metabelian associative algebras

2017

Self Cite

View full text Add to dashboard Cite

show abstract

“…Theorem 2.4. A Leibniz algebra g has the derived algebra g ′ of dimension 1 if and only if g is isomorphic to one of the Leibniz algebras P λ , P (f ) or P (Λ, f ) from the three families defined above, where P = 0 is a vector space, λ ∈ P * \ {0}, f ∈ Bil (P × P, k) \ {0} and (Λ, f ) ∈ (P * \ {0}) × Bil (P × P, k) satisfying (17). Furthermore, we have:…”

Section: On the Structure And The Classification Of Metabelian Leibnimentioning

confidence: 99%

Itô’s theorem and metabelian Leibniz algebras

Agore

Militaru

2014

Linear and Multilinear Algebra

View full text Add to dashboard Cite

We prove that the celebrated Itô's theorem for groups remains valid at the level of Leibniz algebras: if g is a Leibniz algebra such that g = A + B, for two abelian subalgebras A and B, then g is metabelian, i.e. [ [g, g], [g, g] ] = 0. A structure type theorem for metabelian Leibniz/Lie algebras is proved. All metabelian Leibniz algebras having the derived algebra of dimension 1 are described, classified and their automorphisms groups are explicitly determined as subgroups of a semidirect product of groups P * ⋉ k * × Aut k (P ) associated to any vector space P . 2010 Mathematics Subject Classification. 17A32, 17A60, 17B30. Key words and phrases. metabelian Leibniz (Lie) algebras. A.L. Agore is Postdoctoral Fellow of the Fund for Scientific Research Flanders (Belgium) (F.W.O. Vlaanderen).

show abstract

“…Then give an explicit set of generators of the algebra of constants of F δ m (V). Recently, Dangovski et al [3,4] gave some results in this direction. They showed that the algebra of constants F δ m (V) is not finitely generated as an algebra except for some cases, when V is the variety of metabelian Lie algebras, and of metabelian associative algebras.…”

Section: Introductionmentioning

confidence: 98%

The Nowicki conjecture for relatively free algebras

Centrone

Findik

2020

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

A linear locally nilpotent derivation of the polynomial algebra K[Xm] in m variables, over a field K of characteristic 0, is called a Weitzenböck derivation. It is well known from the classical theorem of Weitzenböck that the algebra of constants K[Xm] δ of a Weitzenböck derivation δ is finitely generated. Let m = 2d, and the Weitzenböck derivation δ act on the the polynomial algebra K[X 2d ] in 2d variables as follows:The conjecture was proved by several authors based on different techniques. We apply the same idea to several relatively free algebras of rank 2d. We give the infinite set of generators of the algebra of constants in the the free metabelian associative algebras F 2d (A), and finite set of generators in the free centre-by-metabelian Lie algebras F 2d (C) and the free algebra F 2d (G) in the variety generated by the identities of infinite dimensional Grassmann algebra.2010 Mathematics Subject Classification. 17B01; 17B30; 17B40; 16R10; 16S15;16W25; 13N15; 13A50.Key words and phrases. Free metabelian associative algebras; free centre-by-metabelian Lie algebras; algebras of constants; Weitzenböck derivations, the Nowicki conjecture.

show abstract

Weitzenböck derivations of free metabelian Lie algebras

Cited by 10 publications

References 13 publications

Weitzenböck derivations of free metabelian associative algebras

Weitzenböck derivations of free metabelian associative algebras

Itô’s theorem and metabelian Leibniz algebras

The Nowicki conjecture for relatively free algebras

Contact Info

Product

Resources

About