An Algebro-geometric Construction of Lower Central Series of Associative Algebras

Jordan, David A.; Orem, Hendrik

doi:10.1093/imrn/rnu125

Cited by 11 publications

(34 citation statements)

References 19 publications

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“…It is also established (see [10,13]) that C is a finitely generated algebra over Q and N j a finitely generated module over C. By a standard result from commutative algebra, in order to show that N j is a finite dimensional Q-vector space, we only have to show that N j has finite support as a coherent sheaf on Spec C. Using results from [12], we find that M 2 (A)/M 3 (A) is a nilpotent ideal in C. In other words, Spec X and Spec C share the same reduced scheme (in particular, the same underlying topological space).…”

Section: Finite-dimensionalitymentioning

confidence: 91%

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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: Finite-dimensionalitymentioning

confidence: 91%

“…Let x be a reduced and smooth point of Spec X. Then, from [13,Corollary 3.9] we have N j (A) x = N j (A x ), where the subscript x denotes the completion at x. Moreover, our assumptions on X guarantee that A x is commutative (being Q[[x]] or Q), therefore N j (A) x = 0.…”

Section: Finite-dimensionalitymentioning

confidence: 99%

On lower central series quotients of finitely generated algebras over Z

2015

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“…The remarkable paper of Kapranov [12] puts forward a framework of noncommutative geometry in which the local objects are NC-complete algebras with a smoothness property. Definition 1.2.…”

Section: Introductionmentioning

confidence: 99%

“…4 Any formally smooth (sometimes called quasi-free) algebra is locally free, by the formal tubular neighborhood theorem ( [5], Section 6, Theorem 2). Section 5 of [12] contains non-affine examples:…”

Section: Introductionmentioning

confidence: 99%

Formal geometry for noncommutative manifolds

Orem¹

2015

Journal of Algebra

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This paper develops the tools of formal algebraic geometry in the setting of noncommutative manifolds, roughly ringed spaces locally modeled on the free associative algebra. We define a notion of noncommutative coordinate system, which is a principal bundle for an appropriate group of local coordinate changes. These bundles are shown to carry a natural flat connection with properties analogous to the classical Gelfand-Kazhdan structure. Every noncommutative manifold has an underlying smooth variety given by abelianization. A basic question is existence and uniqueness of noncommutative thickenings of a smooth variety, i.e., finding noncommutative manifolds abelianizing to a given smooth variety. We obtain new results in this direction by showing that noncommutative coordinate systems always arise as reductions of structure group of the commutative bundle of coordinate systems on the underlying smooth variety; this also explains a relationship between D-modules on the commutative variety and sheaves of modules for the noncommutative structure sheaf.

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“…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]).…”

mentioning

confidence: 99%

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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An Algebro-geometric Construction of Lower Central Series of Associative Algebras

Cited by 11 publications

References 19 publications

On lower central series quotients of finitely generated algebras over Z

On lower central series quotients of finitely generated algebras over Z

Formal geometry for noncommutative manifolds

Relations in universal Lie nilpotent associative algebras of class 4

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