Products of commutators in a Lie nilpotent associative algebra

Abstract

Let $F$ be a field and let $F \langle X \rangle$ be the free unital associative algebra over $F$ freely generated by an infinite countable set $X = \{x_1, x_2, \dots \}$. Define a left-normed commutator $[a_1, a_2, \dots, a_n]$ recursively by $[a_1, a_2] = a_1 a_2 - a_2 a_1$, $[a_1, \dots, a_{n-1}, a_n] = [[a_1, \dots, a_{n-1}], a_n]$ $(n \ge 3)$. For $n \ge 2$, let $T^{(n)}$ be the two-sided ideal in $F \langle X \rangle$ generated by all commutators $[a_1, a_2, \dots, a_n]$ ($a_i \in F \langle X \rangle)$. Let $F$ be a field of characteristic $0$. In 2008 Etingof, Kim and Ma conjectured that $T^{(m)} T^{(n)} \subset T^{(m+n -1)}$ if and only if $m$ or $n$ is odd. In 2010 Bapat and Jordan confirmed the "if" direction of the conjecture: if at least one of the numbers $m$, $n$ is odd then $T^{(m)} T^{(n)} \subset T^{(m + n -1)}.$ The aim of the present note is to confirm the "only if" direction of the conjecture. We prove that if $m = 2 m'$ and $n = 2 n'$ are even then $T^{(m)} T^{(n)} \nsubseteq T^{(m +n -1)}.$ Our result is valid over any field $F$.Comment: 8 pages, remarks and references added, typos fixe

Products of several commutators in a Lie nilpotent associative algebra

“…In a particular case when k = 2 and m 1 , m 2 are even Theorem 1.7 has been recently proved by Grishin and Pchelintsev [12] and independently by the authors of the present article [8]. In another particular case when m 1 = m 2 = · · · = m k−1 = 2 and m k is even this theorem has been proved by Grishin, Tsybulya and Shokola [13,Theorem 3].…”

“…The following two lemmas are well known (see, for instance, [14, Lemma 2.1], [15, Example 3.8]); their proofs can also be found in [8]. Now we are in a position to complete the proof of Theorem 1.9.…”

“…Then it is easy to check that G is a nilpotent group of class 2 so (a, b)c = c(a, b) for all a, b, c ∈ G and, therefore, (a, bc) = (a, c)c −1 (a, b)c = (a, b)(a, c) (see [8] for more details). It is clear that the quotient group G/G ′ is an elementary abelian 2-group so…”

“…Remark. Recall that in the proof of Theorem 1.9 we use the same algebra A that was used in the proof of [8,Theorem 1.4]. Note that in both proofs one can choose the algebra A different from one used in our proofs.…”

Os polinômios centrais de algumas álgebras associativas Lie nilpotentes

“…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)…”