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 $[a_1,a_2]=a_1{a}_2-a_2{a}_1$, $[a_1,\dots ,a_{n-1},a_n]=\left[\right[a_1,\dots ,a_{n-1}],a_n](n\ge 3)$. For n≥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∈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∈X.

KEYWORDS:

• Commutator
• Lie nilpotent associative algebra
• polynomial identity
• T-ideal

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16R10
• 16R40

Acknowledgments

The authors thank Plamen Koshlukov for useful suggestions that improve the exposition.