Asymptotics of graded codimension of upper triangular matrices

摘要

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) .