论文标题
分级谎言代数的特性
Specht property of varieties of graded Lie algebras
论文作者
论文摘要
令$ ut_n（f）$为$ n \ times n $ n $上三角矩阵的代数，并表示$ ut_n（f）^{（ - ）} $相对于$ ut_n（f）$的向量空间，相对于通常的blacketator（formutator），在$ ut_n（f）的矢量空间上，换括号（f）。在本文中，我们对$ \ mathbb {z} _n $的理想的SpecHt属性给出了积极的答案，当特征$ p $ of $ p $ of $ f $ is $ f $ is 0或大于$ n-1 $时。也就是说，我们证明，自由分级的谎言代数中的每个级别身份的理想都包含$ ut_n（f）^{（ - ）} $的分级身份，都是有限的。 此外，我们表明，如果$ f $是特征$ p = 2 $的无限字段，则$ \ mathbb {z} _3 $ - 授予$ ut_3^{（ - ）}（f）$不满足Specht属性的身份。更确切地说，我们明确构建了一个渐变身份的理想，其中包含$ ut_3^{（ - ）}（f）$的理想，并且它不是有限地生成的，作为分级身份的理想。
Let $UT_n(F)$ be the algebra of the $n\times n$ upper triangular matrices and denote $UT_n(F)^{(-)}$ the Lie algebra on the vector space of $UT_n(F)$ with respect to the usual bracket (commutator), over an infinite field $F$. In this paper, we give a positive answer to the Specht property for the ideal of the $\mathbb{Z}_n$-graded identities of $UT_n(F)^{(-)}$ with the canonical grading when the characteristic $p$ of $F$ is 0 or is larger than $n-1$. Namely we prove that every ideal of graded identities in the free graded Lie algebra that contains the graded identities of $UT_n(F)^{(-)}$, is finitely based. Moreover we show that if $F$ is an infinite field of characteristic $p=2$ then the $\mathbb{Z}_3$-graded identities of $UT_3^{(-)}(F)$ do not satisfy the Specht property. More precisely, we construct explicitly an ideal of graded identities containing that of $UT_3^{(-)}(F)$, and which is not finitely generated as an ideal of graded identities.
