Asymptotics of graded codimension of upper triangular matrices

作者：Yasumura Felipe Yukihide^*; Koshlukov Plamen Emilov

来源：Israel Journal of Mathematics, 2018, 223(1): 423-439.

DOI：10.1007/s11856-017-1623-7

The graded exponent is an important invariant of group graded PI-algebras. In this paper we study a specific elementary grading on the algebra of upper triangular matrices UT (n) , compute its codimensions, and use this grading to find the asymptotic behaviour of the codimensions of any elementary grading on UT (n) , for any group. Moreover, we extend this to the Lie case, and obtain, for any elementary grading on the Lie algebra UT (n) ((-)), an upper bound and a lower bound for the asymptotic behaviour of its codimensions. Also, we obtain the graded exponent of any grading on UT (n) ((-)) and for any grading on the Jordan algebra UJ (n) .
It turns out that the graded exponent for UT (n) , considered as an associative, Jordan or Lie algebra, for any grading, coincides with the exponent of the ordinary case. In the associative case, the asymptotic behaviour of the codimensions of any grading on UT (n) coincides with the asymptotic behaviour of the ordinary codimensions. But this is not the case for the graded asymptotics of the codimensions of the Lie algebra UT (n) ((-)).

• 出版日期2018-2