Relations in universal Lie nilpotent associative algebras of class 4

Abstract

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...

Products of several commutators in a Lie nilpotent associative algebra

“…Note that v ij appears in (5) if and only if (i, j) ∈ P. Let N = k i=1 m i and let µ : P → {1, 2, . .…”

Os polinômios centrais de algumas álgebras associativas Lie nilpotentes

“…O objetivo desta seção é dar uma descrição da álgebra Q 3 = F X /T (3) . O lema seguinte é bem conhecido, veja por exemplo [2,9,13,18,20,32]. Lema 1.18.…”

“…são multilineares e determinados por seus multi-graus. Como T (3) é multi-homogêneo (Proposição 1.10), basta mostrar que todo polinômio da forma (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) não pertence a T (3)…”

“…Seja g + T (3) = ∑ t α t a t + T (3) onde α t ∈ F e os a t + T (3) são elementos distintos da forma (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17). Observe que os polinômios a t não dependem de x 0 e x 1 (porque g = g(x 2 , .…”

“…Nesta seção exibiremos algumas relações na álgebra F X /T (4) . O Lema seguinte é bem conhecido, veja [9,11,16,21,33,34,44].…”