FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1997, VOLUME 3, NUMBER 2, PAGES 625-630

E. V. Loukoianova (Loukoyanova)

Abstract

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```
For special type of (associative) polynomials
```$f$
and simple algebras $L$
the problem of recognition of identity $f$ in quotient algebra
$U_L/J$ of universal enveloping algebra $U_L$ by arbitrary ideal $J$ ,
where $J$ is given by its generators, is solved.
The central point of the solution is the

\begin{theorem}

Let $l_1, \ldots , l_p$ be Lie (associative)
polynomials with
non-intersecting sets of variables
which are not identities in $L$ ,
$f = \prod\limits_{i=1}^{p}
l_{i}(x_{i_{1}},\ldots, x_{i_{n_{i}}} )$,
then the verbal ideal $T_f(U_L)$ generated by polynomial $f$ in $U_L$
is equal to $U_L{}^p$ .

\end{theorem}

In particular, $U_L/T_f(U_L)$ is a nilpotent
algebra of degree $p$ .

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Last modified: September 30, 2001