FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(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

Theorem. Let $l$₁, ¼ ,l_p be Lie (associative) polynomials with non-intersecting sets of variables which are not identities in $L$, $f\; =$P_i=1^p l_i(x_i1,¼, x_ini), then the verbal ideal $T$_f(U_L) generated by polynomial $f$ in $U$_L is equal to $U$_L^p.

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

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Last modified: September 30, 2001