Relations in universal Lie nilpotent associative algebras of class 4

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 $[a_1,a_2,\dots ,a_n]$ inductively 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 $K⟨X⟩$ generated by all commutators $[a_1,a_2,\dots ,a_n]$ $(a_i\in K⟨X⟩)$. It can be easily seen that the ideal $T^{(2)}$ is generated (as a two-sided ideal in $K⟨X⟩$) by the commutators $[x_1,x_2]$ $(x_i\in X)$. It is well-known that $T^{(3)}$ is generated by the polynomials $[x_1,x_2,x_3]$ and $[x_1,x_2][x_3,x_4]+[x_1,x_3][x_2,x_4]$ $(x_i\in X)$. A similar generating set for $T^{(4)}$ contains 3 types of polynomials in $x_i\in X$ if $\frac{1}{3}\in K$ and 5 types if $\frac{1}{3}\notin K$. In the present article we exhibit a generating set for $T^{(5)}$ that contains 8 types of polynomials in $x_i\in X$.

19 pages. v.2: minor revisions, introduction extended, references added

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• On $[A,A]/[A,[A,A]]$ and on a $W_n$-action on the consecutive commutators of free associative algebra
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Cited by in corpus (6)

• Products of commutators in a Lie nilpotent associative algebra
• The torsion subgroup of the additive group of a Lie nilpotent associative ring of class 3
• On Lower Central Series Quotients of Finitely Generated Algebras over $\mathbb{Z}$
• On some products of commutators in an associative ring
• Products of several commutators in a Lie nilpotent associative algebra
• On Lie nilpotent associative algebras