(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

## Recognition of identities in quotient algebras of universal enveloping algebras

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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