Asymptotics of graded codimension of upper triangular matrices

Author(s)

Yasumura, Felipe Yukihide ^[1] ; Koshlukov, Plamen Emilov ^[1]

Affiliation

^[1] Univ Estadual Campinas, Dept Math, 651 Sergio Buarque de Holanda, BR-13083859 Campinas, SP - Brazil

• Document type: Journal article
• Source: Israel Journal of Mathematics; v. 223, n. 1, p. 423-439, FEB 2018.
• Web of Science Citations: 0

Abstract

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) ((-)). (AU)

• FAPESP's process: 14/09310-5 - Algebraic structures and their representations
• Grantee: Vyacheslav Futorny
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 13/22802-1 - Graded identities in Lie and Jordan algebras
• Grantee: Felipe Yukihide Yasumura
• Support Opportunities: Scholarships in Brazil - Doctorate