arXiv 0710.55682007-10-30math.RA

矩阵代数的分次恒等式与泛分次代数

Graded identities of matrix algebras and the universal graded algebra

Eli Aljadeff, Darrell Haile, Michael Natapov

备注：: 27 pages

摘要：

本文考虑M_n（C）上的细G-分次（即复数上的矩阵代数的分次，其中每个分量是1维的）。提供这种分级的群已知是可解的。我们考虑G-分次恒等式的T-理想，并证明它是由一种特殊类型的二项式恒等式生成的，我们称之为初等恒等式。特别地，我们证明了分次恒等式的理想是作为T-理想生成的。接下来，给出这样的分级，我们构造一个泛代数U_{G,c}在两个不同的方式：一个通过多项式身份和其他一个通过一个通用的两个上循环（这平行的经典结构在非分级的情况下）。我们证明了U_{G,c}的一个合适的中心局部化是Azumaya在其中心上，而且它的同态像恰好是M_n（C）的G-分次形式.最后，我们考虑中心对偶环Q（U_{G,c}）（这是一个F-中心单代数，其中F=Frac（Z），Z是U_{G,c}的中心）。利用作者的一个较早的结果（见E. Aljadeff，D. Haile和M.吴晓波，可除代数的射影基与中心型群，以色列数学杂志146（2005）317-335和M. Natapov arXiv：0710.5468v1 [math.RA]）我们证明了这是一个非常显式（和短）的幂零群族的除代数。因此，对于群G，使得Q（U_{G,c}）不是一个除代数，可以找到一个非单位多项式p（x_{i,g}），使得p（x_{i,g}）^r是某个整数r的分次单位。我们用M_6（C）的精细G-分次来说明这一现象，这里G是S_3和C_6的半直积.

英文摘要：

We consider fine G-gradings on M_n(C) (i.e. gradings of the matrix algebra over the complex numbers where each component is 1 dimensional). Groups which provide such a grading are known to be solvable. We consider the T-ideal of G-graded identities and show that it is generated by a special type of binomial identities which we call elementary. In particular we show that the ideal of graded identities is finitely generated as a T-ideal. Next, given such grading we construct a universal algebra U_{G,c} in two different ways: one by means of polynomial identities and the other one by means of a generic two-cocycle (this parallels the classical constructions in the non-graded case). We show that a suitable central localization of U_{G,c} is Azumaya over its center and moreover, its homomorphic images are precisely the G-graded forms of M_n(C). Finally, we consider the ring of central quotients Q(U_{G,c}) (this is an F-central simple algebra where F=Frac(Z) and Z is the center of of U_{G,c}). Using an earlier results of the authors (see E. Aljadeff, D. Haile and M. Natapov, Projective bases of division algebras and groups of central type, Israel J. Math.146 (2005) 317-335 and M. Natapov arXiv:0710.5468v1 [math.RA]) we show that this is a division algebra for a very explicit (and short) family of nilpotent groups. As a consequence, for groups G such that Q(U_{G,c}) is not a division algebra, one can find a non identity polynomial p(x_{i,g}) such that p(x_{i,g})^r is a graded identity for some integer r. We illustrate this phenomenon with a fine G-grading of M_6(C) where G is a semidirect product of S_3 and C_6.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文