We construct an additive basis for the relatively free associative algebra F (5) ( K ) with the Lie nilpotency identity of degree 5 over an infinite domain K containing $${1 \over 6}$$ 1 6 . We prove that approximately half of the elements in F (5) ( K ) are central. We also prove that the additive group of F (5) (ℤ) lacks the elements of simple degree ≥ 5. We find an asymptotic estimation of the codimension of T-ideal, which is generated by the commutator [ x 1 , x 2 ,…, x 5 ] of degree 5.

中文翻译：

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具有5阶李幂零恒等式的相对自由结合代数的可加基的构造与应用

我们在包含$${1 \over 6}$$ 1 6 的无限域K 上为具有5 次李幂零恒等式的相对自由的结合代数F (5) ( K ) 构建了一个可加基。我们证明 F (5) ( K ) 中大约一半的元素是中心元素。我们还证明了 F (5) (ℤ) 的可加群缺少单次 ≥ 5 的元素。我们找到了 T-ideal 的余维的渐近估计，它由交换子 [ x 1 , x 2 , …, x 5 ] 的阶数 5。