论文标题
分级身份与上三角矩阵代数
Graded identities with involution for the algebra of upper triangular matrices
论文作者
论文摘要
令$ f $为特征零字段。我们证明，如果在$ ut_m（f）$上进行分级的级别承认分级互动，那么该评分是$ \ mathbb {z}^{\ lfloor \ frac {m} {m} {2} {2} \ rfloor} $的级别，并在$ ut_m（f）$上毕业或等同于毕业。 $ ut_m（f）$。 $（\ mathbb {z}^{\ lfloor \ frac {m} {2} {2} \ rfloor}，\ ast）$ - 身份的有限基础，以进行反射和符号互动以及符号互动的增长$（\ Mathbb {z}^{\ lfloor \ frac {m} {2} {2} \ rfloor}，\ ast）$ - codimensions。结果，我们证明，对于$ ut_m（f）$的任何$ g $ - $（g，\ ast）$ - 如果$ m $均匀，$ m $或$ m $或$ m $或$ m+1 $，则$ m $是$ m $。对于代数$ _3（f）$，最多等同于两个非平凡的等级，可以接纳分级相关性：规范$ \ mathbb {z} $ - 分级和$ \ mathbb {z} _2 _2 _2 $ - 级别的分级为$（0,1,1,1,1,0）$。我们确定$（\ Mathbb {z} _2，\ ast）$ - 身份的基础，并证明该指数为$ 3 $。因此，我们得出的结论是，$ ut_3（f）$的普通$ \ ast $ - expent是$ 3 $。
Let $F$ be a field of characteristic zero. We prove that if a group grading on $UT_m(F)$ admits a graded involution then this grading is a coarsening of a $\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor}$-grading on $UT_m(F)$ and the graded involution is equivalent to the reflection or symplectic involution on $UT_m(F)$. A finite basis for the $(\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor},\ast)$-identities is exhibited for the reflection and symplectic involutions and the asymptotic growth of the $(\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor},\ast)$-codimensions is determined. As a consequence we prove that for any $G$-grading on $UT_m(F)$ and any graded involution the $(G,\ast)$-exponent is $m$ if $m$ is even and either $m$ or $m+1$ if $m$ is odd. For the algebra $UT_3(F)$ there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical $\mathbb{Z}$-grading and the $\mathbb{Z}_2$-grading induced by $(0,1,0)$. We determine a basis for the $(\mathbb{Z}_2,\ast)$-identities and prove that the exponent is $3$. Hence we conclude that the ordinary $\ast$-exponent for $UT_3(F)$ is $3$.
