On Lie nilpotent associative algebras
Claud W. G. Dias Jr, Alexei Krasilnikov

TL;DR
This paper investigates conditions under which higher-order Lie commutators vanish in unital associative algebras, establishing a criterion involving generators and their pairwise products over certain rings.
Contribution
It proves a new equivalence relating the vanishing of n-fold commutators in associative algebras to those in generators and their pairwise products, extending known results from groups and Lie algebras.
Findings
Higher-order commutators vanish iff they vanish on generators and pairwise products
The result holds over rings containing 1/3, ensuring certain algebraic properties
Provides a criterion for nilpotency in associative algebras based on generators
Abstract
Let be a group generated by a set . It is well known and easy to check that \[ [g_1, g_2, \dots ,g_n] = 1 \mbox{ for all } g_i \in G \qquad \iff \qquad [x_1, x_2, \dots , x_n] =1 \mbox{ for all } x_i \in X. \] Let be a Lie algebra generated by a set . Then it is also well known and easy to check that \[ [h_1, h_2, \dots , h_n] = 0 \mbox{ for all } h_i \in L \qquad \iff \qquad [x_1, x_2, \dots ,x_n] = 0 \mbox{ for all } x_i \in X. \] Now let be a unital associative algebra generated by a set . Then the assertion similar to the above does not hold: for , it is easy to find an algebra with a generating set such that for all but for some . However, we prove the following result. Let be a unital associative and commutative ring such that . Let …
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Finite Group Theory Research
