Lower central series of a free associative algebra over the integers and finite fields

Bhupatiraju, Surya; Etingof, Pavel; Jordan, David A.; Kuszmaul, William; Li, Jason

doi:10.1016/j.jalgebra.2012.07.052

Cited by 13 publications

(58 citation statements)

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“…It was noticed in [4] that there are no torsion elements in N k (A n (Z)) when k and n are small. A natural guess would be that this holds for every k and n. Surprisingly, Krasilnikov found a 3-torsion in N 3 (A 5 (Z)) and more torsion elements for higher k and n (see [15]).…”

Section: Appendix a Torsion Subgroups In N K (A N (Z Z Z))mentioning

confidence: 99%

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On lower central series quotients of finitely generated algebras over Z

2015

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Let A be an associative unital algebra, B k its successive quotients of lower central series and N k the successive quotients of ideals generated by lower central series. The geometric and algebraic aspects of B k 's and N k 's have been of great interest since the pioneering work of [11]. In this paper, we will concentrate on the case where A is a noncommutative polynomial algebra over Z modulo a single homogeneous relation. Both the torsion part and the free part of B k 's and N k 's are explored. Many examples are demonstrated in detail, and several general theorems are proved. Finally we end up with Appendix A about the torsion subgroups of N k (A n (Z)) and some open problems.

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Section: Appendix a Torsion Subgroups In N K (A N (Z Z Z))mentioning

confidence: 99%

“…On the other hand, Bhupatiraju et al [4] studied the case where A is the free algebra over Z or finite fields. Specifically, they were able to describe almost all the torsion appearing in B 2 (A n (Z)).…”

Section: Introductionmentioning

confidence: 99%

On lower central series quotients of finitely generated algebras over Z

2015

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“…, a n ] (a i ∈ A). The study of these quotients L i /L i+1 was initiated in 2007 in a pioneering article of Feigin and Shoikhet [12] for A = C X ; further results on this subject can be found, for example, in [1,3,4,5,6,7,9,10,11,19,20]. Since T (n) (A) is the ideal in A generated by L n , some results about the quotients T (i) (A)/T (i+1) (A) were obtained in these articles as well; in [7,11,19,20] the latter quotients were the primary objects of study.…”

Section: Introductionmentioning

confidence: 99%

“…It is well-known that T (3) is generated by the polynomials (see, for instance, [6,13,17,23]). If 1 3 ∈ K then a similar generating set for T (4) contains the polynomials of 3 types (see [11,14,24,26]): (1) [ (see [8,Theorem 1.3]).…”

Section: Introductionmentioning

confidence: 99%

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Relations in universal Lie nilpotent associative algebras of class 4

Costa

Krasilnikov

2017

Communications in Algebra

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Abstract. Let K be a unital associative and commutative ring and let K X be the free unital associative K-algebra on a non-empty set X of free generators. Define a left-normed commutator [a1, a2, . . . , an] inductively by [a1, a2] = a1a2 − a2a1, [a1, . . . , an−1, an] = [[a1, . . . , an−1], an] (n ≥ 3). For n ≥ 2, let T (n) be the two-sided ideal in K X generated by all commutators [a1, a2, . . . , an] (ai ∈ K X ).It can be easily seen that the ideal T (2) is generated (as a two-sided ideal in K X ) by the commutators [x1, x2] (xi ∈ X). It is well-known that T (3) is generated by the polynomials [x1, x2, x3] and [x1, x2][x3, x4] + [x1, x3][x2, x4] (xi ∈ X). A similar generating set for T (4) contains 3 types of polynomials in xi ∈ X if 1 3 ∈ K and 5 types if 1 3 / ∈ K. In the present article we exhibit a generating set for T (5) that contains 8 types of polynomials in xi ∈ X. IntroductionLet K be a unital associative and commutative ring and let A be a unital associative K-algebra. Define a left-normed commutator [a 1 , a 2 , . . . , a n ] inductively by [a 1 , a 2 ] = a 1 a 2 −a 2 a 1 , [a 1 , . . . , a n−1 , a n ] = [[a 1 , . . . , a n−1 ], a n ] (n ≥ 3). For n ≥ 2, let T (n) (A) be the two-sided ideal in A generated by all commutators [a 1 , a 2 , . . . , a n ] (a i ∈ A). An algebra A is called Lie nilpotent (of class at most n − 1) if T (n) (A) = 0 for some n ≥ 2.Let K X be the free unital associative K-algebra on a non-empty set X of free generators. Define T (n) = T (n) (K X ). The quotient algebra K X /T (n) can be viewed as the universal Lie nilpotent associative K-algebra of class n − 1 generated by X.The study of Lie nilpotent associative rings and algebras was started by Jennings [18] in 1947. Since then Lie nilpotent associative rings and algebras have been investigated in various papers from various points of view; see, for instance, [2,14,15,16,21,24,25,26] and the bibliography there.Recent interest in Lie nilpotent associative algebras was motivated by the study of the quotients L i /L i+1 of the lower central series of the associated Lie algebra of an associative algebra A; here L n is the linear span in A of the set of all commutators [a 1 , a 2 , . . . , a n ] (a i ∈ A). The study of these quotients L i /L i+1 was initiated in 2007 in a pioneering article of Feigin and Shoikhet [12] for A = C X ; further results on this subject can be found, for example, in [1,3,4,5,6,7,9,10,11,19,20]. Since T (n) (A) is the ideal in A generated by L n , some results about the quotients T (i) (A)/T (i+1) (A) were obtained in these articles as well; in [7,11,19,20] the latter quotients were the primary objects of study.It can be easily seen that the ideal T (2) is generated (as a two-sided ideal in K X ) by the commutators [x 1 , x 2 ] (x i ∈ X). It is well-known that T (3) is generated by the polynomials (see, for instance, [6,13,17,23]). If 1 3 ∈ K then a similar generating set for T (4) contains the polynomials of 3 types (see [11,14,24,26]): (see [8, Theorem 1.3]). In the latter case, in general, the poly...

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The additive group of a Lie nilpotent associative ring

Krasilnikov

2013

Journal of Algebra

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Abstract. Let Z X be the free unitary associative ring freely generated by an infinite countable set X = {x1, x2, . . .(n) be the two-sided ideal in Z X generated by all commutators [a1, a2, . . . , an] (ai ∈ Z X ). It can be easily seen that the additive group of the quotient ring Z X /T (2) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul and Li have noted that the additive group of Z X /T (3) is also free abelian. In the present note we show that this is not the case for Z X /T (4) . More precisely, let T (3,2) be the ideal in Z X generated by T (4) together with all elements [a1, a2, a3][a4, a5] (ai ∈ Z X ). We prove that T (3,2) /T (4) is a non-trivial elementary abelian 3-group and the additive group of Z X /T (3,2) is free abelian.

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Lower central series of a free associative algebra over the integers and finite fields

Cited by 13 publications

References 9 publications

On lower central series quotients of finitely generated algebras over Z

On lower central series quotients of finitely generated algebras over Z

Relations in universal Lie nilpotent associative algebras of class 4

The additive group of a Lie nilpotent associative ring

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