Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Relatively free associative algebras of ranks 2 and 3 with Lie
  nilpotency identity and systems of generators for some T-spaces   We study relatively free associative algebras $F^{(n)}_r$ of ranks $r=2,3$
with the identity $[x_1,\dots, x_n]=0$ of Lie nilpotency of step $n\geqslant 3$
over a field $K$ of characteristic $\neq 2,3$. First we prove a Theorem on the
inclusion $ T^{(m)}T^{(n)}\subseteq T^{(m+n-1)}$ for an associative algebra $A$
of rank $3$, where $T^{(n)}=T^{(n)}(A)$ is a T-ideal of $A$ generated by the
commutator $[x_1,\dots, x_n]$; the restriction on rank is essential. Further,
we describe 3-variable identities of the algebra $F^{(n)}$. In particular, the
obtained description implies that $Z(F^{(n)}_r)=(T^{(n-1)}+Z_q)(F^{(n)}_r)$,
where $p=\mathrm{char}(K)\geqslant 5$ and $q$ is a least power of $p$ such that
$q\geqslant n-1$ and $Z_q$ is a T-space generated by $x^q$. We also prove the
equality $Z(F^{(n)}_2)=F^{(n)}_2\cap Z(F^{(n)})$. Finally, we obtain certain
generalizations and refinements of some results by A. V. Grishin and V. V.
Shchigolev, respectively. For example, we prove that a unital algebra
$F^{(n)}_2$ $(n\geqslant 4)$ over a field $K$ of characteristic $p\geqslant n$
possesses a finite strictly descending "composition" series of T-ideals
$T^{(3)}=T_1\supset T_2\supset \dots \supset T_k\supset T_{k+1}=0$ such that
each quotient $T_i/T_{i+1}$ does not contain any proper T-spaces.
  Key words: Lie nilpotency identity, center, kernel, proper polynomial,
3-variable identity, T-space.

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The categories of this paper are: cmp-lg
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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: math-ph
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: stat
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat